Do I need to obtain a student pilot certificate to transition from Glider to Sport Pilot?
You can't get another student certificate because as a Private Pilot you don't need one (and can't get one). Rather, you're a Private Pilot adding to your existing certificate.

Private pilots already have sport privileges, but if the LSA is a new category/class for you then you still need an endorsement for it. The requirements are in 14 CFR 61.303 (and see AOPA's guide too). You can check the table yourself, but let's assume you're a Private Pilot and you have no medical but you do have a US driver's license. That means the sixth row in the table applies to you:


You must hold the endorsements required in 61.321 for the LSA's category and class
You must comply with the limitations in 61.315, except 61.315(c)(14) and, if a private pilot or higher, 61.315(c)(7)


61.321 requires a logbook endorsement from an instructor who's trained you in the areas listed in 61.309 and 61.311; a proficiency check from a different instructor; an application to the FAA; and a logbook endorsement from the check instructor. You also need a certain number of hours, depending on the category/class of the LSA; see 61.313.

I assume your glider experience already covers a lot of the knowledge and experience required, but your sport instructor will have to determine exactly what training you still need.

(This appears to be the opposite of this question, by the way.)