If they are registered in your state, then I don't believe you will be able to register the same name. If they are in a different state, then I believe you can.
But there are practical reasons why you should want a unique name, if possible.
One reason is the domain name. Do they already have "blahblahblahprinting.com?" If so, you may have to settle for a variation. So if you need a different domain, you may as well use the domain name as the company name.
If their business is already established, they may have good SEO. So if anyone is searching for your company, they may find them instead. They may do different printing, but if a potential customer finds them instead of you, they may choose to find another print shop and you may lose out on a sale.
Another reason is trademarking. See if they have the name registered with the USPTO. If so, you may want to steer clear otherwise you could be infringing on their mark. If not, you could be ok. But since you are not the first to use the name, you may not be able to register it if you ever wish to.
So it may be a good idea to be as unique as possible. It could be something simple as "Blah Blah Blah Screen Printing."