Using apostrophes with plural and possession

"Guest's". From the fact that "palm" and "hand" are singular, you are apparently talking about only one guest. (Well, one guest at a time.) So to make "guest" possessive, we follow the basic rule of adding an apostrophe and an "s".

Yeah, you need an apostrophe, and I'd say it should be "guests'". Generally, one has events for more than one guest, which is why I suggest plural. But the idiom is "in the palm of your hand". Use the singular when everyone being referred to has one (in this case, only one is operative regardless of the fact that most everyone has two). [I'm sure that I read this advice in a Grammar Girl tip.] Otherwise, the sentence is saying something ambiguous: "in the palms of their hands".

You wouldn't want to say "Place your powder on the lesions in your patients' mouths" (each patient has but one mouth and may have only a single lesion in that one mouth) but "Place the tablet on the lesion in a patient's mouth".

So maybe you'd want to say it another way: Bring your event into the palm of each guest's hand.

Not using an apostrophe is probably not unheard of: Mothers Day, Ladies Room, and a few other standard usages argue for no apostrophe in certain cases. But I don't think this is one of those.