If the antibody can recognize hamster epitope, you can use. However the recognition efficacy (hope you understand what i mean) if different between species, therefore you have to use hamster standard instead of mice or rat.
The chances of error will be high if somebody uses different species ELISA kits, better course will be to use the species specific kits. Further there will be very less acceptance of the results obtained form different species kits.
It actually depends on several factors. First, what protein are you trying to detect and quantify? Second, from where the kit is purchased (Company) ? Have you heard about cross species reactivity? For example, if you are trying to detect a protein from hamster using mouse ELISA kit, then you need to consider whether the primary antibody is polyclonal or monoclonal. If the primary antibody is polyclonal, then the chances of detecting your hamster protein will be more. But, if the primary antibody is monoclonal, then the chances will be less. However, a monoclonal antibody (primary) can still recognize the hamster protein if it exhibits cross species reactivity. Finally, as Katsura said, it depends on the epitope that antibody is recognizing. Hope this helps.