Is there a way to question a buyer’s legitimacy?

[inceptions]

If your buyer paid and eBay marked the transaction as ready to ship, then you need to ship. There is no process or reason to investigate your buyer's legitimacy.

What problem or suspicion do you have with the transaction?

[wil81]

I wouldn’t question the buyer’s legitimacy if he paid, the buyer didn’t pay promptly and didn’t respond to 3 inquiries. 

[inceptions]

If a buyer doesn't pay, on day 5 just cancel the transaction using the reason that the buyer didn't pay. He will earn an unpaid item strike on his account and you can move forward.

You can add the buyer to your blocked buyer list here:

https://www.ebay.com/bmgt/buyerrequirements

To protect yourself from repeat offenders, scroll down that page and turn on the option to auto-block buyers with 2 or more unpaid items within 12 months.