Terry, that was my first thought as well: it sounds like a for-profit school. Sounds like Jenni's advisor is trying to hit a quota...and filling out financial aid info is considered a good indicator as to whether she'll enroll.
Jenni, if you are still interested in this school, put the advisor on the spot. Ask her if filling it out will expedite any potential grants or money the school will give you. It won't because they don't give any financial assistance?? That's reason number one they don't need it today.
Now ask her if you give her the info today, when will you get your aid package. Is it well before you plan to enroll? Is it after you plan to enroll? If either answer is "yes", that's reason number two they don't need it today.
Then ask her if you can get admitted without filling out the FAFSA. If she says yes you can get admitted, that's reason number three they don't need it today. Tell her it's your preference to wait until enrolled. If she refuses to give you an answer, let her know that you're not having this trouble with any of the other schools where you're applied. And be firm in saying that you don't like to give out private information about yourself except when necessary. Although in my opinion, if you can't get a straight answer today...how confident that you'll get straight answers while you attend?