There's a shoot of what seems to be Poncirus Trifoliata coming out the base of my Bergamot orange (see picture below). Is it possible to cut and root it? If so, how do I go about it? Thanks in advance.

I would let it get bigger and air layer one of the branches because poncirus is VERY hard to root from cuttings.

Yes, usually you can break it off at the base and just stick it in moist soil. You stand a better chance if you let it harden off first.