• Ability to understand and the willingness to sign a written informed consent document

• Participants must have histologically or cytologically confirmed cutaneous Kaposi sarcoma. Participants must have measurable disease with a minimum of five bi-dimensionally measurable KS cutaneous marker lesions. If fewer than five bi-dimensionally measurable marker lesions are available, the total surface area of the marker lesion(s) must be \>= 700 mm\^2

• Participants must have documentation of HIV status. If HIV negative, documentation of a negative HIV rapid test within 21 days before enrollment. If HIV positive, documentation of HIV-1 infection by means of any one of the following:

‣ Documentation of HIV diagnosis in the medical record by a licensed health care provider

⁃ Documentation of receipt of antiretroviral therapy (ART) (at least two different medications that do not constitute a prescription for pre exposure prophylaxis \[PrEP\]) by a licensed health care provider. Documentation may be a record of an ART prescription in the participant's medical record, a written prescription in the name of the participant for ART, or pill bottles for ART with a label showing the participant's name

⁃ HIV-1 ribonucleic acid (RNA) detection by a licensed HIV-1 RNA assay demonstrating \> 1000 RNA copies/mL

⁃ Any licensed HIV screening antibody and/or HIV antibody/antigen combination assay confirmed by a second licensed HIV assay such as a HIV-1 Western blot confirmation or Multispot HIV-1/HIV-2 Rapid Test or HIV Antibody HIV-1/HIV-2 Differentiation Assay

∙ Note: The term licensed refers to a kit that has been certified or licensed by an oversight body within the participating country and validated internally (e.g., United States \[U.S.\] Food and Drug Administration \[FDA\]). WHO (World Health Organization) and CDC (Centers for Disease Control and Prevention) guidelines mandate that confirmation of the initial test result must use a test that is different from the one used for the initial assessment. A reactive initial rapid test should be confirmed by either another type of rapid assay or an extracellular interactome assay (E/CIA )that is based on a different antigen preparation and/or different test principle (e.g., indirect versus competitive), or a Western blot or a plasma HIV-1 RNA viral load

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 50%)

• Life expectancy of greater than 3 months

• Absolute neutrophil count: \>= 1,000/mm\^3 (within 21 days before enrollment)

• Hemoglobin: \> 8 g/dL (within 21 days before enrollment)

• Platelets: \>= 75,000/mm\^3 (within 21 days before enrollment). Platelet transfusions to help participants meet eligibility criteria are not allowed within 3 days before study enrollment

• Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) (within 21 days before enrollment)

‣ If the elevated bilirubin is felt to be secondary to indinavir or atazanavir therapy, then subjects will be allowed on protocol without any limit on the total bilirubin if the direct bilirubin is normal

• Creatinine:

‣ Serum creatinine levels within normal institutional limits; or, creatinine clearance \>= 30 mL/min (as calculated per the Cockcroft-Gault Equation (within 21 days before enrollment)

• Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification within 6 months before study enrollment. To be eligible for this trial, participants must be class 2B or better within 6 months before enrollment

• Ixazomib can cause fetal harm. For this reason and because proteasome inhibitors as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (barrier method of birth control and another method such as hormone contraception simultaneously; abstinence) before study entry, the duration of study participation, and 90 days after completion of ixazomib administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception or abstain from heterosexual contact before the study, for the duration of study participation, and for 90 days after completion of ixazomib administration

• If HIV positive, participants must have been on antiretroviral therapy (ART) with optimum anti-viral response for at least 12 weeks. Three-drug ART regimens which include a protease inhibitor or a nucleoside reverse transcriptase inhibitor are acceptable, as is the recently FDA-approved injectable ART regimen, cabotegravir with rilpivirine. Additional appropriate regimens in management of HIV as specified in https://clinicalinfo.hiv.gov/ are acceptable as long as they do not include moderate or strong CYP3A4 inducers. Changes of antiretroviral therapy within the prior 12 weeks for toxicity/convenience reasons are allowed (as long as participants are on a stable regimen for 4 weeks per Section 3.1.11). However, if changes in anti-HIV therapy are due to inadequate HIV control and occurred within the 3 months prior to protocol consent, the patient is not eligible until 12 weeks after optimal control is reached.

• If HIV positive, must have been on a STABLE anti-retroviral therapy for at least 4 weeks. The non-nucleoside reverse transcriptase inhibitors (NNRTIs) efavirenz and etravirine are moderate CYP3A4 inducers which will reduce ixazomib exposure and are therefore prohibited. Potential participants requiring change of antiretroviral therapy to avoid moderate to strong CYP3A4 inducers or to pursue better HIV management with an alternate antiretroviral regimen should defer enrollment until completing 4 weeks of the new ART regimen. There should be no intention to change the regimen for the duration of the study.

• HIV positive participants must not show recent improvement in KS on ART alone that may confound response evaluation, within the following parameters:

‣ If on ART 12 to 24 weeks, participants must show a burden of disease requiring further systemic KS-directed treatment (i.e. advanced cutaneous disease, presence of lymphedema or asymptomatic visceral disease, painful lesions) or evidence of KS progression:

⁃ i.e. any new lesion(s);

⁃ spreading of lesions by any measurable degree;

⁃ development of ulceration;

⁃ worsening edema documented by circumferential measure of limb or body;

⁃ increase in symptoms such as pain, including negative psychological impact;

⁃ any degree of disease worsening by imaging that would prompt expert assessment to recommend further systemic treatment with delay

⁃ If on ART for \>24 weeks, must show no evidence of regression in last 8 weeks

• Patients who have residual active KS lesions (not merely tattoo effect) after receipt of recent KS-directed therapy are eligible to participate even if there was no interval progression since completion of therapy as long as: (1) the degree of residual disease merits systemic therapy (i.e. advanced cutaneous disease, presence of lymphedema or asymptomatic visceral disease, painful lesions)); and (2) at least 4 weeks have passed since receipt of the most recent KS-directed therapy.