Here is the PATCH call:

    @HttpPatch
    global static String doPatch(String firstname, String lastname, String email, String phone, String street, String city, String state, String zip, String country) {
    
    RestRequest req = RestContext.request;
    
    // see if a wicell_id was part of the uri
    String sample_id = req.requestURI.substring(req.requestURI.lastIndexOf('/')+1);
    
    if (sample_id != '') {
        Savepoint sp = Database.setSavePoint();
    
        try {
            
            Contact[] contactToUpdate = [SELECT Id FROM Contact WHERE Sample_Id__c =: sample_id];
            if(contactToUpdate.size() > 0){
                for(Contact c : contactToUpdate){
                    c.FirstName = firstname;
                    c.LastName = lastname;
                    c.Email = email;
                    c.Phone = phone;
                    c.MailingStreet = street;
                    c.MailingCity = city;
                    c.MailingState = state;
                    c.MailingPostalCode = zip;
                    c.MailingCountry = country;
                }
                update contactToUpdate;     
            }
            else {
                Lead[] leadToUpdate = [SELECT Id FROM Lead WHERE IsConverted = false AND Sample_Id__c =: sample_id];
                if(leadToUpdate.size() > 0){
                    for(Lead l : leadToUpdate){
                        l.FirstName = firstname;
                        l.LastName = lastname;
                        l.Email = email;
                        l.Phone = phone;
                        l.Street = street;
                        l.City = city;
                        l.State = state;
                        l.PostalCode = zip;
                        l.Country = country;
                    }
                    update leadToUpdate;
                }    
            } 
            return sample_id;
        }
        catch (DMLException e1) {
            Database.rollback(sp);
            return e1.getDMLMessage(0);
        }
        catch (Exception e2) {
            Database.rollback(sp);
            return e2.getMessage();
        }
    }
    else {
        return 'Invalid operation';
    }
  }

Below is the latest test code for the PATCH call. Note, I'm getting the following error: List has no rows for assignment to SObject on the following line about half way through the below code: String orgId = [SELECT Id from Account LIMIT 1].Id;

I have @isTest listed at the top of my class.

    static testMethod void testdoPatch() {
 
        Lead l = new Lead();
    
        l.firstname = 'Wain';
        l.lastname = 'Rolen';
        l.company = 'Test Company';
        l.email = 'Wain@test.com';
        l.phone = '6884382997';
        l.street = '123 Test Ave';
        l.city = 'Somewhere';
        l.state = 'MA';
        l.PostalCode = '87945';
        l.country = 'US';
        insert(l);
        String sample_Id = l.Sample_Id__c;
        
        String JSONMsg = JSON.serialize(l);
        
        RestRequest req = new RestRequest();
        req.addHeader('Content-Type', 'application/json'); // Add a JSON Header as it is validated
        req.requestURI = 'https://cs4.salesforce.com/services/apexrest/v.1/leads/' + sample_Id;
        req.httpMethod = 'PATCH';
        
        req.requestBody = Blob.valueof(JSONMsg); // Add JSON Message as a POST
        
        RestResponse res = new RestResponse();
        RestContext.request = req;
        RestContext.response = res;
        LeadRestSvc.doPatch('Wain','Rolen','Wain@test.com','6884382997','12345 Test Ave','Somewhere','MA','87945','US'); 
        
        //Test doPatch for contacts
        String orgId = [SELECT Id from Account LIMIT 1].Id;
        
        Contact c = new Contact();
    
        c.firstname = 'Wain';
        c.lastname = 'Rolen';
        c.AccountId = orgId;
        c.email = 'Wain@test.com';
        c.phone = '6884382997';
        c.mailingstreet = '123 Test Ave';
        c.mailingcity = 'Somewhere';
        c.mailingstate = 'MA';
        c.mailingPostalCode = '87945';
        c.mailingcountry = 'US';
        c.Sample_Id__c = sample_Id;
        insert(c);
        String contactSampleId = c.Sample_Id__c;
        
        String JSONMsg2 = JSON.serialize(c);
        
        RestRequest req2 = new RestRequest();
        req2.addHeader('Content-Type', 'application/json'); // Add a JSON Header as it is validated
        req2.requestURI = 'https://cs4.salesforce.com/services/apexrest/v.1/leads/' + contactSampleId;
        req2.httpMethod = 'PATCH';
        
        req2.requestBody = Blob.valueof(JSONMsg2); // Add JSON Message as a POST
        
        RestResponse res2 = new RestResponse();
        RestContext.request = req2;
        RestContext.response = res2;
        LeadRestSvc.doPatch('Wain','Rolen','Wain@test.com','6884382997','12345 Test Ave','Somewhere','MA','87945','US');
    }

I tried to mark your earlier response as the answer but I wasn't given the option. Maybe this is due to a different post being marked as the answer by another user?

Thank you for your help!!!