Suppose that $X(n)=(X_1(n),X_2(n))\in\mathbb{N}^2$ is a discrete time homogeneous Markov chain with uniformly bounded jumps (see below). Assume that the chain is ergodic and let $\pi$ be its invariant probability measure.
If $Y$ is random variable distributed as $\pi$, is it true that $\mathbb{E}[Y]<\infty$? In other words, I'm interested in understanding whether ergodicity and uniformly bounded jumps ensure a finite first moment.
Uniformly bounded jumps. I mean that there exists some finite $c>0$ such that the transition $x\mapsto x+(\Delta_1,\Delta_2)$ can occur only if $-c\le \Delta_1,\Delta_2<c$, for all $x$.
Even in one dimension, the answer is no: you can cook up an ergodic Markov chain on $\mathbb N$ with an arbitrary limiting distribution, even if you require jumps to be bounded by $\pm1$. You ask a question about 2D chains, but we can extend a 1D example to 2D in a number of ways, such as:
For an example of how to attain an arbitrary probability distribution, take the following Markov chain. Here, each state loops back to itself with whatever the leftover probability is, and $C$ is chosen to make sure that that leftover probability is positive ($C = \frac12$ will do).
This Markov chain has limiting distribution $\pi_n = \frac6{\pi^2 n^2}$ because this satisfies the detailed balance equations $\pi_n p_{n,n+1} = \pi_{n+1} p_{n+1,n}$ for all $n$. So here, the expected value is $$\sum_{n\ge 1}n \pi_n = \sum_{n \ge 1} \frac{6}{\pi^2n} = \infty.$$