Have a swiching load which has current peaks larger than power supply can output. Would putting a cap such as this https://www.digikey.com/product-detail/en/united-chemi-con/E32D500HPN104MDA5M/565-3298-ND/2095912 across the switching load resolve the issue?

Would the repetitive inrush (from power cycles - powering and shutting off power to the circuit) into the 100mF cap destroy the power supply?

 

What is a "Switching" load ?
PWM ?

We need a schematic diagram.

Usually capacitors are added to the output of the Power Supply, not "across the switching load"
Many SMPS will become unstable when driving large capacitors.
You may need to limit the turn-on inrush.