Should I provide a gap certificate for NIT Jalandhar?

Yes, the gap certificate is required for admission. Don't be concerned about these secondary documents when applying. In the institute, you can give an undertaking for those documents. Students are typically given a 15-day time period, which can be extended.

These colleges are government institutions and operate in the same manner. Simply give them what they are requesting, and you will be able to obtain this document locally, wherever you are studying. There is no requirement that it be from your home state.