I'm not sure how I need to proceed on this and can use a bit of advice.

I have a non-profit's web site that wishes to allow their members to
fill out information regarding summer camp information (name, contact
info, mailing info, physician's name, etc) online and submit it. I can
handle taking the info with PHP and putting it into a mySQL database easily.

But what has me stuck is that they want the ability to print out hard
copies of selected submissions (their choice) if they need to.

Would I have to display all the database entries in a form on a new page
with a select box next to each entry and have them pick that way, then
loop through and print out the selected ones or is there a better, more
direct (read: easier) way to accomplish this?

Does this sound like something for which there may be a pre-written PHP
script that can be altered?

You list the entries and add sth. like
<input type="select" name="subm[]" value="$ID" />
after each one where $ID is the submission's id.

Now you can simply evaluate the form data:
foreach ($_POST['subm'] as $id) {
// print submission no. $id here
}