Fast question!

Yes. I applied for mine last month after being laid off at my last job and before starting my AZ course. I don't have the AZ licence yet (thanks, Drive Test strikers!), but my FAST card arrived in the mail a week ago.

On the application form, it will ask you which port of entry you will be using the most. I wrote in "unknown at this time" and it wasn't an issue for them. What you have to make sure of is that you account for ALL of your work history and former addresses for the required time frames. Even if you were unemployed, you still need to write down the dates of your unemployment or it will get sent back for correction.