They do actually.
It’s an interesting question so I decided to look it up and put some numbers to it – according to [this paper](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7552655/), just shy of 20% of the (farmed) foxes they studied were carrying Salmonella. In the discussion they cite several other papers that found incidences of ~5% or less in wild foxes.
There are lots and LOTS of different subspecies and serovars of Salmonella. Certain strains are [specially adapted](https://www.sciencedirect.com/science/article/pii/S0378113508001855?via=ihub) to survive the low pH of carnivores guts.
Animals, birds and reptiles especially, are significant reservoirs of all kinds of Salmonella strains. The only kind of Salmonella animals (other than non-human primates) *can’t* suffer from is Typhoid, as that’s human and primate specific.
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