Summary:Yes, lizards drink water. The idea that they absorb water through their skin seems to have been over-turned.
Some lizards drink water by lapping it into their mouths:
The kinematics of drinking of Lacerta viridis were analysed. A drinking bout is composed of four phases: approach, immersion, emersion and withdrawal. The tongue and gravity are central to moving water through successive compartments of the buccal cavity and into the oesophagus. Upon the basis of formifunction analysis of water intake and transport, a kinematic model of drinking in lizards is proposed.
Some lizards drink water by having it soak into their mouths, via the skin:
When wetted, animals assumed a stereotyped posture [...] Drinking involved repeated, slight tongue protrusion, but without lapping from the substrate. [...] He hypothesize that the drinking posture 1) facilitates drinking rain water, and/or 2) facilitates drinking water condensed on the skin and moved to the mouth by capillary action.
Some lizards collect the rain water that lands on their backs, drinking it through their mouths:
SEM stereophotographs illustrate the interscalar channels through which water is carried, apparently by capillary action, over body surfaces to the jaws.
The idea that lizards absorb water into their body through their skin appears to have fallen into disfavour:
In 1923 Davey was the first to report his observations on the remarkable hygroscopic nature of thorny devil skin, presumably with the animal drinking through the walls of its body. He reported that its skin “absorbs moisture almost as readily as blotting paper.”
This unlikely scenario for a desert lizard, having skin permeable to water (water is more likely to move out than in), was not formally questioned until 1962.
Two scientists at the University of Western Australia, in Perth, used a colloidal blue dye to show that water actually moves up and across the surfaces of moloch skin very rapidly in all directions, including up and onto the head and back. It did not penetrate the skin surface. The dyed water from the skin’s surface was found in a lizard’s stomach after the animal, standing in dyed water, slowly and slightly opened and closed its jaws.
In 1982, the water-transport hypothesis was again revised. Researchers now noted that the water carried over the skin by capillary action actually did not move across the surfaces of scales, where it may be initially captured, but it moved down channels between individual scales, subsequently flowing through this channel network toward the mouth for ingestion.
(That doesn't prove that there isn't any single lizard that does it.)
Here is another more readable description of how lizards drink.