So, lets say I do marketing for Unicorn Vendors. There are unicorn vendors in every city in the US. I'm tasked with creating websites for my clients. Will it hurt my local SEO rankings if I make a generic website that applies to all unicorn vendors, but use different domains/logos/header and title text for each client?
Short answer. Depends. If these sites appear to be different sites with duplicate content, then Yes! It will hurt.
Changing the domain name, title, and logo is not enough.
Google and Bing read the site content rather thoroughly and will find duplicate content and even uses AI analysis to find very similar content which appears to be duplicate content changed to fool the search engines.
Following your scenario, I would create a single website with each customer as either a sub-directory or a sub-domain. I would use markup for the customer pages to indicate that these are vendors and indicate address, phone numbers, and so forth. If a customer has their own website, then link to their site from the vendor pages.