As far as I know there is no MOVR token on BSC yet - so it won’t be possible to directly get MOVR. Wanchain is working on this but it’s not ready yet, if they are live you could buy wanMOVR and bridge those to Moonriver Chain directly.
You need to bridge the BNB and swap the BNB for MOVR.
However it’s true you need MOVR for Gas, so far you have two options:
Transfer some MOVR from one of the exchanges (Kucoin, Kraken or Gate.io)