No, it doesn't sound very natural here. 'Exigencies' suggests compelling or unexpected needs, not just the need for a software system. You could say "for your needs" but it might be better to take it out entirely, given that most any system is used for its users' needs.
Also, in formal writing you should generally spell out acronyms the first time they're used. Also there's a missing article "a". Also, as written the sentence has an incorrect comma splice. Correcting these, you would have:
Skint is a bring-your-own-device (BYOD) system, which means that you are supposed to bring your own laptop in order to use Skint.
EDIT: I had also deleted the second "system". It's not ideal to repeat "system", but it might be necessary if (as commented) the word Skint is used for other things and could be ambiguous. Proper nouns like "Skint" generally don't take an article, but "system" does, so you can either "use Skint" or "use the system" or "use the Skint system":
Skint is a bring-your-own-device (BYOD) system, which means that you are supposed to bring your own laptop in order to use the system.
Skint is a bring-your-own-device (BYOD) system, which means that you are supposed to bring your own laptop in order to use the Skint system.