What role does salt play in pumpkin pie?

Salt is fundamental to our sense of taste, leaving it out will definitely affect the flavor of a pie (or a cake, or a steak, or whatever) negatively. However, leaving it out shouldn't have affected the texture, you would need to use much more salt than I imagine your pie filling recipe called for to affect the texture at all.

tl;dr - Maybe salt was responsible for your texture problem, but it's iffy.

Salt is usually thought of as a flavoring agent only, but salt does some serious jobs in the chemistry of cooking. It's worth looking at some options for what this awesome rock does to your custard (and being a cooked mixture of dairy, sugar and eggs, pumpkin pie filling is a custard).

Without any details about how the texture was off, we're forced to guess what your unintentionally low-sodium recipe may have done to your pie. (I would note that I would expect the flavor to be a bit bland or uninspiring without salt before I would expect the structural issues I'll describe below.)

Potential Number 1: Salt's Impact on Boiling and Steam

This all comes down to one word: leavening. As water turns to steam, it can become trapped as bubbles in your food and increase volume. Salt is known to increase boiling point so the water stays liquid longer in the oven. If salt is to impact your leavening, a higher boiling point will mean your pie is denser and wetter, because less water escapes as steam.

How much? Well, exact numbers are hard, so let's hand waive some lazy math! With the water in your pumpkin and the water in your sweetened condensed milk, let's say that you have a pint of water in your filling. Assuming your recipe calls for a half teaspoon of salt (2.5 grams) that will give you approximately a .0944M NaCl solution. Which will raise the boiling point of this water from 100*C to 100.097*C. (EXAMPLE 13.9.3 shows how this formula works.)

A tenth of a degree increase is underwhelming. Especially when you pair this with the fact that eggs cook significantly lower temperature than water boils (about 60*-70*C). The custards start to sets at about 70*C. Far before there will be any significant leavening from steam. Generally speaking, never take your custards to 80*C, because that's criminal overcooking and I will call the food police on you.

Potential Number 2: Salt's Impact on Protein Coagulation

While some custards use starches like corn starch or tapioca as thickeners, a decent portion of the thickening is provided by heat making the proteins (Ovalbumin) in the egg coagulate. So, how does salt impact coagulation? It turns out that it's kind of complicated. At different concentrations it can either help OR hurt coagulation.

Even though it's thick with the addition of pumpkin, I (rightly or wrongly) think of pumpkin pie as a diluted custard which lends itself to this study or ovalbumin coagulation. Long, dry science writing short? Low concentrations of salt helps proteins coagulate when they are so diluted they normally couldn't coagulate.

So yes, salt looks like it will help solidify the eggs in your pumpkin pie. How much? Well, I don't have a way to test the gel strength in pumpkin pies, but comparing finished cooked pie versus the thickness of the pumpkin you scooped from the can, yeah. It's a bit more solid, but not a huge amount and who knows how much of that is pectin from the pumpkin?

Was your pie softer or looser than normal? Was it more like the pumpkin from the can texture than pie? This might lead to that result.

Possible, but I'm going to doubt it. I think it's much more likely that it was either slightly under/over done or under/over mixed. Probably look to technique before the forgotten salt.

Added salt can dissolve in, and alter the boiling point of, any water in your pie filling. This could have an effect on how much moisture is retained in the baking, process which may effect the texture.