My Dad carried a Win 94 in .32 Special for years. Dropped at least 3 deer I know of. It will work just fine.
There are, however, two drawbacks to the .32 Special. One real, one potential. The potential is that .32Specials have a reputation for losing their accuracy (sometimes) with no apparent reason. Not saying it will, only that it can.
The main, and the real drawback to the .32 Special is that ammo production is, and has for some time been "seasonal".
"Seasonal" ammo production does not mean it is produced every season. What it means is that it is produced every few YEARS, give or take. When the inventory reaches a certain low point, the factory sets up the production line, does a run to bring the inventory back up to the desired level, then packs up the tooling and puts it away, until the next time. Depending on the sales of the ammo, it might be a year, two, or 6.
IF there is a "run" on the ammo (unexpected buying surge) which sometimes happens due to a popular movie, or other reason for renewed interest, there can be a "dry spell" with no "new" ammo available, until the next factory run.
If you like and want the .32 Special, buy several years worth of ammo, when you have the chance, and you'll likely always have some.
If you are a handloader, don't sweat it!
Components are always out there, and if you can't find .32 Special brass you can make acceptable cases from the common .30-30 very easily.