Yes, a PC-lycanthrope can multiattack. Under any interpretation some of them do, and I argue that all do. (And you can skip to the last paragraph if you don't need to see the argument.)
Like you said, The PC-lycanthrope gains all of the abilities that don't involve equipment. A very strict reading would say that any action-paragraph that mentions equipment isn't imported: under this reading the wereboar and wererat import Multiattack just fine, but the other lycanthropes all don't import Multiattack.
This reading is too strict to make sense. It has the effect of making some afflicted with lycanthropy faster and able to multiattack (e.g. wereboar), but some not (e.g. werebear). This, even though all monsters with lycanthropy can multiattack.
Contrast this with a reading in which each portion of an action-paragraph is evaluated against "involves equipment."1
The effect of this reading is that the PC-lycanthrope gains natural attacks and the ability to multiattack, but doesn't gain any equipment.
A few examples, all quotes from the 5e MM pp.208-11:
A wereboar gains Tusks, doesn't gain Maul, and Multiattack reads as-is:
Multiattack. The wereboar makes two attacks, only one of which can be with its tusks.
A werebear gains Bite, Claw, doesn't gain Greataxe, and Multiattack can be read like:
Multiattack. In bear form, the werebear makes two claw attacks. In humanoid form, it makes two
greataxe attacks. In hybrid form it can attack like a bear or a humanoid.
A werewolf gains Bite, Claws, doesn't gain Spear, and Multiattack reads like:
Multiattack (Humanoid or Hybrid form only). The werewolf makes two attacks: one with its bite and one with its claws
But even this is a little too much lawyering for me: note that the wereboar or werebear both get to make multiattacks that involve weapons, but the werewolf doesn't. Why? Because we're over-thinking it.
The action is the bold-italic word or few words: the paragraph that follows just explains it. (See also AngryGM's article on abbreviating stat-blocks.2)
Under this philosophy a PC-wereboar gains Tusks and Multiattack. A PC-werebear gains Bite, Claw, and Multiattack. And a PC-werewolf gains Bite, Claws, and Multiattack. I then understand that wereboar can attack with tusks and a weapon, but not twice with tusks. Werewolves can attack twice--one bite and one that somehow uses the hands--but the PC didn't gain a were-spear when afflicted.
1 - Note that for all of the lycanthropes in the MM, the only actions that do mention equipment are those relating to their (artificial) weapon attacks.
2 - Be warned that AngryGM's posts are liberally sprinkled with profanity and/or rude language. Nothing you might not hear on television, but still worth knowing before you innocently head over there.