The physical admit card with the verification sticker is not mandatory for JEE Main counselling.  A soft copy will suffice for most purposes.

JEE Main counselling is conducted online through the Joint Seat Allocation Authority (JoSAA) portal. The verification of documents primarily happens virtually during the college seat allocation process.
The list of documents required for counselling typically includes your JEE Main scorecard, class 10th and 12th mark sheets, photo ID, and category certificate (if applicable). The admit card isn't usually on the mandatory list.

https://engineering.careers360.com/download/ebooks/jee-main-2022-preparation-tips

I hopeit helps!