With a payer swaptions delta and gamma is there a method for approximating pnl for a given move in underlying swap rate? (An equivalent to the Taylor expansion for a vanilla call)
Yes, the Taylor expansion usually works well as the first approximation to explain the P&L when the curve doesn't move a lot. The "delta" (first derivative) is the sensitivity to the parallel shift of the swap curve. The "gamma" (second derivative) is the convexity.
You can get even better P&L explanation by
including separate sensitivities to swap rates at different tenors (most of your sensitivity is to the rate for the maturity of your underlying), rather than assuming parallel shift.
using more sophisticated IR gamma. Depending on your needs, you can use one number risk-weighted to ficus on yourt underlying's maturity. Or you can have a matrix with an entry for each tenor pair.
include the cross-gammas between rates, implied volatility, and time.