Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

There are rigorous constructions of QFTs in infinite volume. Glimm & Jaffe's book does this for interacting 2d scalars (with the assumption that the interactions are not too strong). I'm sure you can find other examples in the literature (or perhaps someone else will point you to them).

However, restricting yourself to a lattice doesn't really buy you much. For one thing, if the lattice action you're using is approximately local and a good approximation to the true effective action, you're probably not too far from the continuum limit anyways.

One of constructive QFT's little surprises -- at least if you were brought up on Peskin & Schroder -- is that removing IR cutoffs is a considerably harder problem than removing UV cutoffs. For one thing, you usually can't just take a limit of the finite-volume measures. Instead, you have to find some collection of observables whose expectation values remain well-defined in the IR limit, and then use something like Minlos' Theorem to infer the existence of a measure. Finding the right observables isn't easy. You want to show that the expection values obey some form of cluster decomposition, so that you can safely ignore things that are happening far away. This is somewhat difficult just in massive 2d scalar field theory, where the correlation functions of the basic observables decay exponentially. (It takes Glimm & Jaffe only a few pages to show the existence of a finite volume continuum limit, but it takes them a few chapters to show the infinite volume limit exists.) It's even harder if your correlators only decay like polynomials. And in theories like Yang-Mills theory or massless 2d scalars, where the correlation functions of basic observables can actually grow with distance, it can become monstrously hard. You have to find just the right observables --e.g., exponentiated fields in the 2d scalar case -- and show that the divergences cancel out. (The Millenium Prize for Yang-Mills theory really amounts to solving the IR problem. The UV problem in finite volume is believed to be basically tractable.)

Constructing the infinite volume lattice theory is not that difficult usually. One typically uses the so-called Griffiths' inequalities. See this article by Sokal (p. 327), as well as this article by Guerra, Rosen and Simon (Section V). In the case of Yang-Mills on a lattice, the infinite volume limit is more difficult. However, it has been done rigorously at large coupling, see this article by Osterwalder and Seiler. I should add that the results I mentioned are for any number of dimensions $d\ge 2$. This is because one is just talking about the infinite volume limit on a fixed lattice. One needs to worry about dimension when one wants to also take the lattice mesh to zero.

One of the most complete books on the lattice infinite volume limit from a mathematically rigorous standpoint is "Gibbs Measures and Phase Transitions" 2nd Edition by H.-O. Georgii. See in particular the notes on sections 4.3 and 4.4 on page 458 which discusses various techniques one can use for unbounded spin systems.