Will only let one submission go through

Try deactivating all other plugins and switching to the default theme.

The green Success message and not receiving any error messages is normally a very clear indication that CF7 is sending the email.

So your 2nd email could be being seen as spam somewhere in the email process. This is usually due to email headers (in particular the From email address used) or email content..

Emails can be considered spam for a whole host of reasons and get rejected at many points in the email process in addition to your spam folder.

See Contact Form 7 Email Issues – this links to a much more comprehensive article which also covers emails being considered spam.

See also Best Practice to Set Up Mail??.

No, it’s not seeing it as spam

Obviously, CF7 views my history and sees I sent a form already, so it won’t allow a second submission.

If a real form user wants to send two forms one after another (which can happen) their second message will not give an error but will not be sent.

As I said, if I send one, it works and the second time I try, it will not give an error message but will not be sent. However, if I then, without logging out, go incongnito on Chrome it suddenly works.

Thank you for your help; I really appreciate it.