No help from this website so I instead contacted a consulting company to look at my code and deebug logs. The code seems to be correct according to them: the problem is likely on the Vendor's side.

In case anyone is interested, I've included the most recent version of code (not finalized version, of course) below:

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public class CatalystPatientInsert {
    
    public static HttpResponse makePostCallout() {
        
        //Help for this class found at "https://salesforce.stackexchange.com/questions/11015/basic-http-authentication"
        String username = 'USERNAME';
        String password = 'PASSWORD';
        Boolean redirect = false;
        
        Http http = new Http();
        HttpRequest request = new HttpRequest();
        String url = 'https://sandbox.vendor.com/api/students/';
        request.setEndpoint(url);
        request.setMethod('POST');
        Blob headerValue = Blob.valueOf(username +':'+password);
        String authorizationHeader = 'Basic ' + EncodingUtil.base64Encode(headerValue);
        system.debug(authorizationHeader);
        request.setHeader('User-Agent','Hopebridge');
        request.setHeader('Content-Type', 'application/json;charset-UTF-8');
        request.setHeader('Authentication', authorizationHeader);
        
        //Help for this class found at "https://developer.salesforce.com/forums/?id=906F0000000BQ30IAG"
        JSONHelpWrapper jwrap = new JSONHelpWrapper('7FBBDC6A-8206-442D-B06D-5724C40CE46B', '7FBBDC6A-8206-442D-B06D-5724C40CE46B', 'JJ',
                               'Test2', null, '2014-03-26T10:05:44', '2014-03-26T10:05:44',
                               'm', true, '2014-03-26T10:05:44', null,
                               'test, jj', 'Autism Spectrum Disorder', false,
                               'America/New_York');
        
        String requestValue = JSON.serialize(jwrap);
        System.debug(requestValue);
        request.setBody(requestValue);
        HttpResponse response = http.send(request);
        // Parse the JSON response
        if (response.getStatusCode() >= 300 && response.getStatusCode() <= 307 && response.getStatusCode() != 306) {
            do {
                redirect = false; // reset the value each time
                String loc = response.getHeader('Location'); // get location of the redirect
                System.debug('The status code returned was not expected: ' + response.getStatusCode() + ' ' + response.getStatus());
                System.debug(response.getHeader('Location'));
                if(loc == null) {
                    redirect = false;
                    continue;
                }
                request.setBody(requestValue);
                request = new HttpRequest();
                request.setEndpoint(loc);
                request.setMethod('POST');
                request.setBody(requestValue);
                response = http.send(request);
                if(response.getStatusCode() != 500) {
                    if(response.getStatusCode() >= 300 && response.getStatusCode() <= 307 && response.getStatusCode() != 306) {
                        System.debug(response.getHeader('Location'));
                        redirect = true;
                    }
                }
                
                
            } while (redirect && Limits.getCallouts() != Limits.getLimitCallouts());
            
        }
        
        if (response.getStatusCode() != 201) {
            
            System.debug('The status code returned was not expected: ' + response.getStatusCode() + ' ' + response.getStatus());
            System.debug(response.getHeader('Location'));
            
        }
        
        System.debug(response.getBody());
        return response;
        
    }
    
}

---------------------------

public class JSONHelpWrapper {
    
    Public String BCBA {get;set;}
    Public String LeadTherapistId {get;set;}
    Public String FirstName {get;set;}
    Public String LastName {get;set;}
    Public String SiteId {get;set;}
    Public String DateOfBirth {get;set;}
    Public String EntryDate {get;set;}
    Public String Gender {get;set;}
    Public Boolean Active {get;set;}
    Public String DateLastUpdated {get;set;}
    Public String ExternalId {get;set;}
    Public String StudentCode {get;set;}
    Public String Diagnosis {get;set;}
    Public Boolean IsUsingTeachingSessions {get;set;}
    Public String TimeZone {get;set;}
    
    Public CatalystWrapper(String BCBA, String LeadTherapistId, String FirstName,
                               String LastName, String SiteId, String DateOfBirth, String EntryDate,
                               String Gender, Boolean Active, String DateLastUpdated, String ExternalId,
                               String StudentCode, String Diagnosis, Boolean IsUsingTeachingSessions,
                               String TimeZone) {
                                   This.BCBA = '7FBBDC6A-8206-442D-B06D-5724C40CE46B';
                                   This.LeadTherapistId = '7FBBDC6A-8206-442D-B06D-5724C40CE46B';
                                   This.FirstName = 'JJ';
                                   This.LastName = 'Test2';
                                   This.SiteId = null;
                                   This.DateOfBirth = '2014-03-26T10:05:44';
                                   This.EntryDate = '2014-03-26T10:05:44';
                                   This.Gender = 'm';
                                   This.Active = true;
                                   This.DateLastUpdated = '2014-03-26T10:05:44';
                                   This.ExternalId = null;
                                   This.StudentCode = 'test, jj';
                                   This.Diagnosis = 'Autism Spectrum Disorder';
                                   This.IsUsingTeachingSessions = false;
                                   This.TimeZone = 'America/New_York';
                               }
    
}