Yes. A non-retriggerable monostable is one thing that will do that (see below also).
Here's a revamp of my earlier circuit, which will trigger the counter within about 1mS from pressing a button.
R1 and C3 provide additional filtering of the supply to reduce the likelihood of spurious resetting of the counter by interference.
R1/C1 and R5/C4 provide switch debounce and prolong the ground pulse for ~100mS, as per your question.
R10/D3/D4 perform a diode-OR function, eliminating the need for a CD4001 as used previously.
