If you entered them under your email account (it's that or "phone" that are your choices when entering contacts), they're on Google Contacts and the next phone you put on that email address will download them.
If not, that's one of the services a good local shop may provide - contact transfer or backup even with no screen.
Otherwise - get the screen replaced by someone local who won't wipe your phone before he starts working on it (which is the normal practice, so you can't accuse him of stealing any data from it).