I just thought I should mention coconut oil as another substitute for fats that are solid at room temperature. I prefer it to the other alternatives mentioned because it is vegetarian, and much less processed then shortening.
It should be able to work in your dough much the same way as shortening - though its temperature range is a bit narrower than shortening, it will be very hard during the winter but can turn mushy (or, occasionally, outright liquid) in summer, depending on its storage. You might wish to treat it like a more forgiving butter and chill it before use, rather than a meltier shortening.
Yet another possible alternative might, maybe, possibly be used is red palm oil. I have seen it in a few stores, next to the coconut oil. As I have not used it myself, I can't attest to its use in baking... but I have seen jars that look like they might be solid at that temperature, and it is described as "semi-solid". Again, chilling may make it workable if it is only a little too liquid at room temperature.
I also thought of ghee, since that's also semisolid at room temp (and can be chilled harder) - but if you're avoiding butter, you may not find that suitable, though you might find it useful to check, as the flavors are not identical.