It depends on how far you want to validate the postal codes from France. If you only want to validate the format, the form field should only check if the input consists of 5 numbers and does not start with a 0. An example of this can be seen here: https://rgxdb.com/r/354H8M0X
To include this in a form plugin of your choice, you’ll need to write this individually with JavaScript and/or PHP. There is a possibility in HTML to add a validation to an input field. Examples can be seen here: https://www.html5pattern.com/Postal_Codes – for France you would have to use the regular expression from the link above. To be able to do this, you would have to modify the form plugin you have chosen individually.
As far as I could research just now, there are no ready plugins for France for such a case. For Contact Form 7 you could add a custom validation as described in the tutorial here: https://contactform7.com/2015/03/28/custom-validation/
If you want to check not only the format but also the actual existence of a postal code, you would first need a complete list of postal codes in France. I don’t know where you could get it – at least in Germany there are such lists at OpenStreetMap or similar open source projects. You could use this list as a basis for such a validation. For Gravity Forms, there is a tutorial on how to validate an input in a form field against such a list: https://gravitywiz.com/gravity-forms-zip-code-validation/
If you need development support, you can certainly find someone here: https://jobs.wordpress.net
If you have further questions about a specific form plugin, contact other support. The commercial plugins can certainly support you with individual developments for an additional charge.
