Study of mRNA-4359 Administered Alone and in Combination With Immune Checkpoint Blockade in Participants With Advanced Solid Tumors

Key Inclusion Criteria: * Males or females ≥18 years of age who have provided written informed consent prior to completing any study-specific procedure. For Arm 2d, participants ≥12 years are eligible with informed consent/assent. * Dose Escalation (Arm 1a): Participant has histologically confirmed locally advanced or metastatic cancer (cutaneous melanoma, NSCLC, non-muscle invasive bladder cancer, head and neck squamous cell carcinoma, microsatellite stable colorectal cancer \[MSS CRC\], basal cell carcinoma, or triple negative breast cancer) with measurable disease as determined by RECIST v1.1. Arm 1a participants must have received, and then progressed, relapsed, or been intolerant to, or ineligible for, at least 1 standard treatment regimen in the advanced or metastatic setting. Participants with a known driver mutation must have also received or been offered a mutation-directed therapy, where indicated. Participants must have a tumor lesion amenable to biopsy and must have another lesion that can be followed for response. * Dose Confirmation (Arm 1b): Participant has histologically confirmed locally advanced or metastatic, and CPI refractory melanoma or locally advanced or metastatic, and CPI refractory NSCLC with measurable disease as determined by RECIST v1.1 who has disease progression after, at least 1 line of standard therapy (no limit to prior lines of therapy), and has been treated with or refused standard of care treatment. Participants in PD arm Group 2 must also have PD-L1 TPS ≥1%. Participants must have primary refractory or acquired secondary resistance to prior immune checkpoint treatments. Primary refractory is defined as prior exposure to anti-programmed death-1 (PD-1)/PD-L1 antibody for at least 6 weeks but no more than 6 months with demonstration of progression on 2 separate scans at least 4 weeks apart but no more than 12 weeks apart and progression occurring within 6 months after first dose of anti-PD-1 antibody. Acquired secondary resistance must have confirmed objective response or prolonged stable disease (SD) (\>6 months), followed by disease progression in the setting of ongoing treatment and confirmed progression on scans at least 4 weeks apart. Participants must have a tumor lesion amenable to biopsy and must have another lesion that can be followed for response. 1. For NSCLC participants with known EGFR, ALK, proto-oncogene tyrosine-protein kinase reactive oxygen species (ROS1), or other actionable mutations for which there are approved targeted therapies, participants must have received prior approved targeted therapy or have been offered and declined approved targeted therapy. 2. Expansion of the melanoma Arm 1b cohort (up to approximately 16 additional evaluable participants) will require prospective central testing of Screening tumor biopsies to confirm PD-L1 TPS ≥1% for eligibility. * Dose Expansion Arms (Arm 2): Participant has histologically confirmed: 1. Arm 2a: Locally advanced or metastatic melanoma who have not yet received any prior systemic therapy for their melanoma in this setting. 2. Arm 2b: Newly diagnosed locally advanced or metastatic NSCLC with a PD-L1 TPS of ≥50% with no known EGFR or ALK positive tumor mutations who have not yet received any prior systemic therapy for their NSCLC (that is, treatment-naive). 3. Arm 2c: Locally advanced or metastatic melanoma in participants who have not received prior systemic therapy for melanoma in the advanced/metastatic setting. 4. Arm 2d: Advanced/metastatic melanoma that is CPI refractory and having a centrally confirmed PD-L1 TPS of ≥1% on their Screening tumor biopsy. * All participants must have measurable disease as determined by RECIST v1.1. * Participants must have a tumor lesion amenable to biopsy and must provide tumor biopsy sample at baseline (archival formalin-fixed, paraffin-embedded \[FFPE\] tissue collected within 90 days of informed consent is accepted as long as no intervening therapy is received, during this time), and optionally at all on-treatment timepoints (including response or progression) if medically feasible. Participants in Arm 2c: Sufficient tumor tissue (slides or FFPE block) for PD-L1 testing is required as per Laboratory Manual. Participants in Arm 2c may be replaced in this cohort if not PD-L1 evaluable. Participants in Arm 1b melanoma expansion cohort and Arm 2d: Central confirmation of PD-L1 TPS score is required prior to enrollment. * If the participant is undergoing a new biopsy, they must have another lesion outside of the lesion biopsied at baseline that can be followed as a RECIST v1.1 target lesion for response. * Participant has an Eastern Cooperative Oncology Group (ECOG) performance status of ≤1. * Participant has adequate hematological and biological function. Key Exclusion Criteria: * Participant has active central nervous system tumors or metastases. * Participant has received treatment with prohibited medications (that is, concurrent anticancer therapy including other chemotherapy, radiation \[local radiation for palliative care is permitted with approval from the Sponsor\], hormonal anticancer treatment, biologic therapy, or immunotherapy) or investigational agents within 5 half-lives or 14 days prior to the first day of study treatment, whichever is shorter. Prior exposure to any investigational or approved agent designed to simultaneously target both indoleamine 2,3-dioxygenase (IDO1) and PD-L1 is also exclusionary. * Participant has required the use of additional immunosuppression (for example, infliximab) other than corticosteroids for the management of an AE, has experienced recurrence of an AE if rechallenged, and currently requires maintenance doses of \>10 milligrams (mg) prednisone or equivalent per day. * Participant has any plan to receive a live attenuated vaccine during study treatment or has received a live vaccine within 30 days before the first dose of study treatment. Examples of live vaccines include, but are not l