A 100A MOSFET is more what you want - the current rating is the current the device
can handle with infinite heatsink at absolute maximum power dissipation - totally
You choose a MOSFET by the on-resistance, Vds(on), such that the power dissipation
is within the bounds of your heatsinking arrangement and the max Vds(on) is much
smaller than Vgs(on).
For a 12V 12.5A load I’d choose a device with <= 0.005 ohms on resistance, limiting
dissipation to under a watt - a medium/small heatsink will handle that nicely. Because of the
high power I’d choose a MOSFET driver chip so that PWM will be efficient and allow use of
a standard MOSFET (logic level MOSFETs are less abundant). The MIC442x family are
available in DIP and SOIC8 and will drive almost any size of MOSFET without the slightest
struggle (good decoupling is absolutely mandatory). There are many many other low-side
driver chips (literally 100’s of product lines).
These days its not too hard to find 20V and 30V MOSFETs down to 0.001 ohms on-resistance,
so a heatsink is not actually necessary.
Dissipation = I^2 x R = 12.5^2 * 0.005 = 0.78W
Vds(on) = I x R = 12.5 * 0.005 = 62 mV
Mosfets don’t have enough power
I suspect you haven’t found a real electronics stockist’s site! Try this one as an example: