Can a unit be sent out to base with charm if she's at Villemaw Lair?

No, a unit cannot be moved from Vilemaw's Lair to a base using Charm.

According to RiftJudge FAQ #6856 and FAQ #4475, Vilemaw's Lair prevents all forms of movement from that battlefield to a base, including movement caused by card effects like Charm.

While you can legally target a unit in Vilemaw's Lair with Charm (as the unit is a valid target), the movement portion of the effect will fail to resolve because the battlefield's "can't" restriction overrides the "can" effect of the spell.