nlab presents a proof that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since such a product would be a topological vector space, by the universal property of the projection maps, and it would also be a product in $\mathbb{R}\text{Vect}$. It is known that every locally compact Hausdorff topological vector space is finite dimensional, see here for example.

However since linear maps in $\mathbb{R}\text{Vect}$ are not in general proper, such a proof doesn't work in the case that you restrict to proper continuous maps between locally compact Hausdorff spaces.

I wanted to know if there is still a known counterexample that means that this category will not have products?

One reason that suggests to me they might is that topological spaces with local homeomorphisms (étale maps) has products, and they are very weird; for example $\mathbb{R}\times\mathbb{R}^2=∅$ in that category. It also massively restricts what maps can be onto compact spaces (they must come from a compact space, which is another 'intuition' for why it might have products.