This is an accepted solution.
Nick here from Shopify. Great question!
To answer it quickly, nothing would particularly happen if you removed the vendor's name from your store. Vendor names are useful when you’re looking to filter products in your admin in various ways. Some store owners like to be able to find their products using different variables using filters such as vendor, inventory, tags, and others which are explained in further detail here. If you find this field serves no purpose for your business it’s certainly not necessary to fill out. I've seen many store owners who decide not to use it, but many who do as well, it would be completely up to you and your preferences. Depending on your preferences you can choose to display the vendors on product pages for customers to see but by default, vendors are turned off from your product pages.
I hope this helps and answers your question.
All the best, Nick