Luton South and South Bedfordshire Labour MP Rachel Hopkins said she had heard "various reasons" for the delays.
"It was pushed from one budget period to the next budget period, then it was Covid, then it was all the designs, then it was 'we're trying to start, but we found some problems'," she said.
She said Network Rail "isn't project-managing it properly enough".
"For them to keep delaying - I don't think it's acceptable."
Station operator Govia Thameslink said it wanted lifts installed "as soon as possible" and was supporting Network Rail.
Station staff, it added, would help passengers and arrange alternative transport to Luton Airport Parkway for those unable to manage stairs.
Since Access for All launched, step-free accessible routes have been built at more than 250 stations.
Gavin Crook, principal programme sponsor for Network Rail's East Midlands route, said: "We are committed to providing access improvements at Luton station and are sorry this is taking longer that we would like."
A DfT spokesperson said: "Everyone should be able to travel with ease and confidence, which is why this government is committed to improving the accessibility of our railways.
"While previous plans for improvements at Luton station were delayed at the time due to design challenges, Network Rail is now responsible for the works, which are progressing."