My impression as a non-native learner is that отказаться (от) would suggest a stronger refusal (perhaps the offer was offensive in some way), while отклонить could mean that the offer was simply uninteresting to you. Is this correct, Helenej?
I think the Russian translation that Helenej suggests would usually mean "they'll keep walking" (or, if you're talking about buses and trains, "They'll keep operating on this route").
But "to keep going" can have other idiomatic senses in English, such as "to be persistent" or "to stay in business" or "to survive," or "to not quit," etc. So depending on the sense, you could use infinitives other than идти after the verb продолжать/продолжить ("to continue").
Like, "The women's shoe-store will keep going" (even though it had to lay off one-third of its employees) could be Магазин продолжит продавать женскую обувь. ("The store will continue to sell women's footwear.")
In other contexts, using some negated construction such as Они не перестанут ("they will not stop...") would sound better than Они продолжат ("they will continue...").
But in any case, "to keep going" is so idiomatic and non-specific that more context is necessary to translate it properly.