Is it possible to see the data of a post request in Firefox or Chrome?

You could just use the Chrome Developer Tools, if you only need to track requests. Activate them with Ctrl+Shift+I and select the Network tab.

This works also when Chrome talks HTTPS with another server (and unless you have the HTTPS private key you cannot use Wireshark to sniff that traffic).

(I copied this answer from this related query.)

With Firefox you can use the Network tab (Ctrl+Shift+E or Command+Option+E). The sub-tab "Params" shows the submitted form data.

Reference: https://developer.mozilla.org/en-US/docs/Tools/Network_Monitor/request_details#Params

Alternatively, in the console (Ctrl+Shift+K or Command+Option+K) right click on the big pane and check "Log Request and Response Bodies". Then when the form is submitted, a line with POST <url> will appear. Click on it; it will open a new window with the form data.

As of the time of originally writing this reply, both methods messed up newlines in textarea fields. The former deleted them, the latter converted them to blanks. I haven't checked with a newer version.