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Gravity is the curvature of spacetime, and its effects travel at lightspeed. However, space is expanding; eventually, light from distant galaxies will become more and more redshifted, and we will no longer be able to see them (source).

As such, there is a limit to how far we could ever possibly see, since light too far will never reach us due to the rapid expansion of space...or at least, if I'm understanding this correctly.

Now, gravitational waves travel at lightspeed. So, after enough time, when an object's light no longer reaches us, will its gravity no longer affect us either?

A better rephrasing is: at a certain point in time, will the gravity of any extremely distant object — even the most massive stars, black holes, or galaxies — simply not affect us whatsoever, in the slightest?

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The answer here is very similar to if you were asking about light.

In principle gravitational waves might allow us to fractions of a second after the big bang. Electromagnetic waves can see back to where the cosmic background radiation formed, about 400,000 years after the big bang.

You are right, the universe has expanded. At the present epoch it is estimated that the observable universe, containing objects that have emitted light or GWs that may reach us now, is around 46 billion light years.

However, it seems quite likely that the universe continues well beyond this horizon, and sources beyond this horizon can never have emitted light or GWs that will reach us.

As the relevant section of wikipedia (https://en.m.wikipedia.org/wiki/Observable_universe) points out, the detection of GWs extends our view very slightly. We cannot "see" with electromagnetic waves beyond 45.7 billion light years because of the "fog" of the cosmic microwave background, but GWs can penetrate this fog allowing us (in principle) to see signals from objects currently 46.6 billion light years away.

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