Gst registration

Yes the supplier is required to obtain GST registration in both states.

Yes they may register with Rajasthan

The GST registration to be taken only in the State where the taxable supplies are provided. The exempted turover to be considered for calculating the aggregate turnover for obtaining the registration. however the requirement to take registration in the Spefic state would araise only if they provide the taxable supplies and where the aggregate turnover exceedes Rs. 20lakhs
Therefore the registration in Rajasthan is not required which provides only exempt supplies.

Since he is engaged in only exempted supplies from Rajasthan, he need not taking any registration for Rajasthan