It's actually a series of simple operations, so you won't have any trouble finding someone.
Who you go with with be largely determined by what the finished bookmarks are supposed to be and look like, and how many you're doing. You'll want to talk to both full-service local printers (not Staples) and screen printers in or near your area. Nothing against Staples; just that independent shops are more flexible and have a bigger range of tools for something like this.
The printer can do something simple like print the bookmarks digitally a few to a sheet (ask to see printed samples first and don't use them if you don't like the image quality).
After that, the sheets can be laminated in glossy or matte film, left alone, or whatever. Matte laminate gives a silky, elegant finish to the product, so look into it. Then the sheets are cut to yield the nearly-finished bookmarks. Some printers have corner rounding devices as well, and nearly all have drills to punch a hole in the bookmark so you can run some yarn or a chain through them. If you wanted to get really fancy you could have a custom die made to punch the bookmarks out of the sheets, but it isn't strictly necessary.
If you're thinking of doing a few thousand bookmarks, it would make sense to run them on a offset printing press.
A silk-screener, meanwhile, can print the bookmarks on a material like plastic vinyl, which you would probably recognize from something like a thicker version of a report cover. Again, this might be appropriate if you're doing zillions of them, but ask anyway because bookmarks are the kind of thing they would likely have a pre-made die for, which would save you lots of money.
If you are referring to silk screen printing.... there is no standard size...