Unfortunately – or fortunately, depending on the point of view) – vergessen is not a dative verb.
You could check this list (be sure to look at part one and part two, which is linked below the table on the first page) whether you spot a verb that you could have confused with vergessen.
But „Sie wird mich vergessen.“ is the correct translation.
It's not in the singular or plural form, agreeing with any person -- it's in the infinitive form (the base form, dictionary form).
As in English, where "forget" is in the base form and not in the form "forgets" that would correspond to "she".
The inflected verb, the one that agrees with the subject, is wird.
(The wir and sie, Sie forms look the same as the infinitive except in the verb "to be", though, so you can't tell those apart just by looking at them. The ihr form is different, though, so there's not "a plural form" of a verb which is used for all plural subjects.)
Sie werden mich nicht vergessen lassen. They will not let me forget. I am confused as to how this sentence works . . . how does one know that "me" is the one not forgetting, and not "they"? OOooh! Perhaps this is why: "Sie werden sich mich nicht vergessen lassen." They will not let themselves forget me." ???
Yeees... but the sentence is confusing and I had to think about it for moment before I realised that it probably means what you intend.
It'd be clearer so say Sie werden es nicht zulassen, dass sie mich vergessen "They will not permit forgetting me" (or literally, They will not permit it, that they forget me).