The lawmakers are reportedly calling for the sack of the Chairman, Committee on House Services, Baballe-Ila, for failing to ensure the delivery of the vehicles ordered since 2015.
The vehicles, numbering 360, are worth about N6.1 billion.
The lawmakers get the official vehicles, better known as "utility or committee" cars every four years.
It was gathered that the House contracted Peugeot Automobile Nigeria Limited to supply 360 units of Peugeot 508 series late in 2015 but only 175 members cars have been supplied so far.
On Thursday, July 27, before the members went on recess, they were said to have demanded explanations from the Speaker, Yakubu Dogara on why the supply is taking forever.
It was learnt that they are especially blaming the Chairman, Committee on House Services, Nasiru Baballe-Ila, for the delay.
Punch quoted a National Assembly source as saying; "Right from when this arrangement started, there was an undertaking that the cars would be supplied in batches.
"More than two years into a four-year tenure, only 175 members have been supplied the cars. Others are still paying from their pockets to work and the suffering continues.
"Members are tired and they can no longer keep calm. It was learnt that the hiccups were caused by what the House later discovered to be ‘Peugeot’s lack of capacity’ to deliver the cars because of the huge number.
"It has got to a pathetic situation whereby members scramble for the few units that are delivered.
"As a matter of fact, they have been compelled to lobby the House Services Committee chairm Toyota Land Cruiser an so that they will be among the lucky ones to get the cars."
Another source added that, "It was the initial plan to go for either Toyota Land Cruiser Prado or Camry, which could be easily delivered.
"But it was Baballe-Ila who used his position as chairman to foist Peugeot on members. Now, Peugeot is unable to deliver the cars two years on.
"Matters came to a head last week just before members went on break as they asked Dogara to sack Baballe-Ila."
Reports said each of the Peugeot vehicle costs N17 million.