Paper statements are in fact becoming kind of old fashioned these days.

You may want to check with your EHR software vendor - many of them will handle your statements for you and charge a flat fee per statement. All you usually have to do is upload your list of statement and they take care of the rest. It's usually cheaper than doing it yourself since they can take advantage of bulk rate mailings.
But if you want to do it in-house, rather than getting a printing vendor, you can probably configure your system to generate envelopes and/or mailing labels. And the statements usually don't even require a printed envelope as they're formatted to fit into a window envelope - just order the correct size window envelopes (both for the outgoing statement and the return envelope) from a Staples or Office Depot type of place, print your statements and fold them the right way so the addresses display. No printing required other than the statements themselves.
Of course, these days even all that is becoming unnecessary. There's software available to just send statement to patients electronically that they can pay online through a portal or by calling an automated phone system, so you don't even have to be involved in it at all. Maybe something to look into that could save your practice both time and money.