The elderly woman can be seen in a picture posted on social media sitting in her wheelchair unattended.
Another passenger said she found the woman at a boarding gate alone.
She was reportedly left at 19:00 for an Easyjet flight which was delayed until 22:00 on Sunday.
A representative from OmniServ said: "We are grateful that this has been brought to our attention. We are taking this matter seriously and investigating to find out exactly what happened here."
An Easyjet spokeswoman said: "Easyjet is very concerned to see this picture and is urgently taking this up with Edinburgh Airport and their special assistance provider Omniserv.
"This is clearly not the customer service and care we expect to be provided to our passengers.
"As such, we have immediately launched an investigation with Omniserv to establish what happened and ensure it cannot happen again."
A spokesman for Edinburgh Airport said: "This is unacceptable. The security and wellbeing of all of our passengers is our number one priority and we are assisting easyJet and Omniserv in their investigation in order to get to the bottom of this and make sure it doesn't happen again."
A firm employed at Edinburgh Airport to give special assistance to reduced mobility passengers has launched an investigation after a grandmother was reportedly left on her own for hours.