USA Swimming's Online Meet Entry (OME) is their own proprietary system and, to our knowledge, does not interface with anyone's
system. So if you entered your swimmers on their site there is no way
to import those entries into your TeamUnify site. We hope one day to be
able to tie into it but can't until they decide to open it up.
Since you enter the swimmers on their site, you may want to avoid the duplication on your own site as far as individual event selection and simply offer people the chance to say yes or no to attending the whole meet, or even just provide a link to the event on USA's site for signup.
If you want to bill for the events, you will need to manually create invoice items for each swimmer.