You can start a sentence with 'among', but it doesn't work in your sentence. Among is used for countable things, and 'equipment' isn't countable. 'on' isn't a good option. Try 'about', 'regarding' or possibly 'for'. You have used 'stops' with plural 'controls', which is incorrect. It's probably more natural to use a superlative with 'Among'. Among our different items of security equipment, we most often get requests about remote controls that stop working.

"Apart from request for our security equipment we also get requests to repair remote controls that stop working" Among is not used like as in your sentence, it is used to say something was is mixed in with or surrounded by something. Or forms a part of something. Requests for our security equipment is one of many among our vast range of services, that includes repair of malfunctioning remote controls. Among is not the best word to use in your sentence.

The sentence is true and sounds quite natural.

Among our different security equipment, we usually get requests on remote controls that stop working.