Things have changed a little bit, but not much...
You'll fill out another
STS-33 application, along with an
SP-66 Consent for Mental Health Records. You'll also need to log in to the NJSP's website, and request an SBI background check. First, call your local PD, and get their 9-digit ORI number (NJ0xxxx00). If they ask you what for, tell them it's needed to submit an electronic request for a State Police background check.
(click the links above to download the forms from NJSP's website. Remember to print two copies of each).
Then, log in to this website:
https://www.njportal.com/njsp/criminalrecords/ApplicationRequest/RequestApplication.aspx and submit your request. It will cost you $20, payable by credit card. Print out the confirmation, and bring that, your completed STS-33, and your completed (but not yet signed) SP-66. You have to sign the SP-66 in the presence of a witness, so wait until you get to the PD to do so. Give them those three things, and you should be all set.
Note - on the electronic form, if you're asked for your SBI number, that's the number on your FID card.
ETA: Yes, you can apply for more than one. Fill in the amount of permits you want in the top-right corner of the STS-33 form. Realistically, you can apply for up to 6 permits at once - as they're good for 90 days, and can be renewed for an additional 90 days. Since you can acquire one handgun per 30 day period, you can fit 6 purchases within those 180 days. Remember you'll owe $2 per permit at the time you pick them up.