The 34-year-old's contract has been terminated by the English governing body after fresh allegations about his behaviour came to light.
In a statement, it was claimed that Sampson committed "inappropriate and unacceptable" behaviour during his time at Bristol Academy, where he worked before becoming the national team coach in 2013.
An investigation into safeguarding allegations in 2014 found that the Welshman "did not pose a risk" working in football, but the FA has decided to take action after learning of the accusations just this week.
Crouch has raised concerns over the governing body's due diligence in its hiring process.
"This situation is a mess and raises very serious questions about whether the historic processes that the FA had in place around the recruitment of coaches were appropriate, for something like this to have been missed," said Crouch.
"The FA are right to have taken action, but reassurance is needed to make sure this does not happen again at any level of coaching."
The termination is not linked to allegations from Eni Aluko, which resulted in Sampson being cleared of wrongdoing in two separate investigations.