Comparing Radiation Plus Cetuximab to Radiation Plus Chemotherapy in People With Head and Neck Cancer Who Cannot Receive Cisplatin

Inclusion Criteria: * Patients must have pathologically confirmed, previously untreated, unresected squamous cell carcinoma of the larynx, hypopharynx, oropharynx, or oral cavity * Local evaluation of p16 status is required for all oropharynx patients prior to registration * Local evaluation of p16 status is recommended for non-oropharynx patients prior to registration * Locoregionally advanced head and neck squamous cell carcinoma (HNSCC) defined as: * Non-oropharynx and p16-negative oropharynx cancer: American Joint Commission on Cancer (AJCC) 8th edition stage III-IVB * Laryngeal, Hypopharyngeal, Oral Cavity, and p16-Negative Oropharyngeal Primaries: * AJCC 8th Edition TNM: T3-4b N0 M0 AJCC 8th Edition Stage: III-IVB * AJCC 8th Edition TNM: T1-4b N1-3 M0 AJCC 8th Edition Stage: III-IVB * p16-positive oropharynx cancer: AJCC 8th edition stage III and selected stage I-II based on smoking status in pack-years * Eligible p16-Positive Oropharyngeal Primaries * AJCC 8th Edition TNM: T1-2 N1 M0 AJCC 8th Edition Stage: I Pack-Years: \> 10 * AJCC 8th Edition TNM: T1-2 N2 M0 AJCC 8th Edition Stage: II Pack-Years: any * AJCC 8th Edition TNM: T3 N0-1 M0 AJCC 8th Edition Stage: II Pack-Years: \> 10 * AJCC 8th Edition TNM: T3 N2 M0 AJCC 8th Edition Stage: II Pack-Years: any * AJCC 8th Edition TNM: T1-3 N3 M0 AJCC 8th Edition Stage: III Pack-Years: any * AJCC 8th Edition TNM: T4 N0-3 M0 AJCC 8th Edition Stage: III Pack-Years: any * Note: Number of pack-years = \[Frequency of smoking (number of cigarettes per day) × duration of cigarette smoking (years)\] / 20 * Note: Cigar and pipe tobacco consumption is not included in calculating the lifetime pack-years. Marijuana consumption is likewise not considered in this calculation. There is also no clear scientific evidence regarding the role of chewing tobacco-containing products in oropharyngeal cancer, although this is possibly more concerning given the proximity of the oral cavity and oropharynx. In any case, investigators should not count use of non-cigarette tobacco products in the pack-years calculation * The following are required prior to registration: * Imaging of the head and neck with a neck CT or MRI (with contrast, unless contraindicated) or PET/CT which includes diagnostic-quality CT of the neck (with contrast, unless contraindicated) * Chest imaging: Chest CT (with contrast, unless contraindicated) or PET/CT * Age ≥ 18 * Complete the online tool at www.nrgoncology.org prior to registration and record the (modified) Charleston Comorbidity Index (CCI), Head and Neck Cancer Intergroup (HNCIG) omega, and G-8 scores on the registration form in Oncology Patient Enrollment Network (OPEN) * Patients must have a contraindication to cisplatin as defined in the following bullet points: * Absolute or relative contraindication to cisplatin, defined as ONE OR MORE of the following prior to registration: * Creatinine clearance (CrCl) \< 60 mL/min by the Cockroft-Gault formula * Pre-existing peripheral (sensory or motor) neuropathy grade ≥ 2 (per Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\] 5.0) * History of hearing loss, defined as either: * Existing need of a hearing aid OR * ≥ 25 decibel shift over 2 contiguous frequencies on a pretreatment hearing test as clinically indicated OR * age ≥ 70 with Head and Neck Cancer Intergroup (HNCIG) omega score \< 0.80 prior to registration OR * Age \< 70 with ALL of the following conditions prior to registration (see Appendix II for calculation instructions): * HNCIG omega score \< 0.80 * (Modified) Charlson Comorbidity Index (CCI) ≥ 1 * G-8 score ≤ 14 * Not pregnant and not nursing * Participants must be able to safely receive the radiation and drug regimens per current Food and Drug Administration (FDA)-approved package insert(s), treating investigator's discretion, and institutional guidelines * No prior systemic therapy for the study cancer; note that prior systemic therapy for a different cancer is allowable * No prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields * No prior surgery for the study cancer