The site, ran by Express Test, is based in the fast car park in front of the terminal.
It went live on Wednesday, with staff and passengers able to pay for a test with results due the next day.
Express Test are now asking customers to email them rather than book through their website, claiming that they are struggling with technical issues due to high demand.
One user told the Edinburgh Evening News: “I booked my test and you are meant to receive an email confirming with a QR code.
"The money came out my account but I didn’t get an email from them.
"Contacting them was impossible, I couldn’t get through and they never answered my email.
"It took two days for me to get the confirmation.”
A coronavirus test from the Edinburgh Airport site costs a whopping £99.
The Express Test Twitter feed has also been the focus of many potential customers voicing their concerns at booking their test.
One user posted a photograph of their phone log, showing they had called the Express Test number 97 times.
Another claimed: “Site crashing already.
"Don’t inspire much confidence if you can’t get the site ready for bookings.
"Also can’t get through on the phone either.
"So will you be able to get results back in time? Loosing confidence before even starting.”
Someone else wrote: “Still no confirmation / booked 48 hours ago / money taken out of my account / no one answering the phones / or responding to emails where I’ve requested a refund.”
A spokesperson from Express Test said: “Due to extremely high demand, we have been experiencing some technical issues with our website and call centre.
"Whilst we are working to resolve this problem we are temporarily asking customers to book a test by emailing [email protected], where our team is working hard to help everyone.
"Anyone who has booked a test will receive a booking confirmation shortly.
"We apologise for any inconvenience.”