Vaccine Therapy With or Without Cyclophosphamide in Treating Patients With Recurrent or Refractory Multiple Myeloma

Inclusion Criteria:

* Myeloma relapsing from partial response or better

  * Patients relapsing \> 18 months from transplant if not on maintenance, or
  * If off maintenance, discontinued at least 6 months ago, or
  * If relapsing on maintenance, at least 3 years from transplant, or
  * Off prior myeloma therapy at least 6 months ago
  * Sufficient tumor burden that is assessable for response

    * Serum M-spike \>= 0.5 g/dL, or
    * If immunoglobulin A (IgA) myeloma, IgA \> 1000 mg/dL, or
    * Difference between involved and uninvolved free light chain (dFLC) \> 10 mg/dL, or
    * Urine M-spike \>= 200 mg/24 hours, or
    * Bone marrow plasmacytosis \>= 10%, or
    * Plasmacytoma \>= 2 cm in diameter
* Absolute neutrophil count (ANC) \>= 1000/uL
* Platelets (PLT) \>= 50,000/uL
* Hemoglobin \>= 8.5 g/dl
* Aspartate aminotransferase (AST) =\< 2 times upper limit of normal
* Creatinine \< 2 times upper limit of normal
* Total bilirubin =\< 1.5 x upper limit of normal
* International normalized ratio (INR) =\< 1.4 x ULN at the time of registration
* Ability to provide informed consent
* Willingness to return to Mayo Clinic Rochester for follow-up
* Life expectancy \>= 12 weeks
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, or 2
* Willingness to provide all biological specimens as required by the protocol
* Negative serum pregnancy test done =\< 7 days prior to registration for women of childbearing potential only
* Measles antibody titer on the BioRad Multiplex assay less than or equal to 1.0

Exclusion Criteria:

* Uncontrolled infection
* Active tuberculosis
* Any myeloma directed therapy within 12 weeks of registration including plasmapheresis or transfusion
* New York Heart Association classification III or IV, known symptomatic coronary artery disease, or symptoms of coronary artery disease on systems review
* Active central nervous system (CNS) disorder or seizure disorder
* Human immunodeficiency virus (HIV) positive test result
* Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (used for a non-Food and Drug Administration \[FDA\] approved indication and in the context of a research investigation)
* Previous exposure to heat inactivated measles virus vaccine (this vaccine was given to some individuals between the years of 1963-1967)
* Any of the following:

  * Pregnant women or women of reproductive ability who are unwilling to use effective contraception
  * Nursing women
  * Men who are unwilling to use a condom (even if they have undergone a prior vasectomy) while having intercourse with any woman, while taking the drug and for 4 weeks after stopping treatment
* Evidence of chronic or acute graft versus host disease or on-going treatment for graft versus host disease from prior allogeneic stem cell transplantation
* Exposure to household contacts =\< 15 months old or household contact with known immunodeficiency