No. You can browse the Play Store in any browser except Firefox for Android. You cannot browse App Store with any browser (also you cannot run an alternative browser engine on iOS)
As one commenter said, you can use it in Chrome on android. And until recently (it's in development now) it was mozzila's decision not to make Firefox for iOS.
What would it mean to make Firefox for iOS? My understanding was that iOS requires all browsers to use the same rendering engine under the hood, so "Firefox" there wouldn't be much more than a different skin for Safari. (Has that changed?)