You don't need a free hand for a spell or ability with a range of "touch" (unless it says otherwise)
The paladin's Lay on Hands feature says:
Your blessed touch can heal wounds. [...] As an action, you can touch a creature and draw power from the pool to restore a number of hit points to that creature, up to the maximum amount remaining in your pool.
It doesn't state any restrictions on how you must touch them, or whether you need a free hand to do so. The feature's name does not necessary impose a restriction on how it works.
Similarly, the spellcasting rules say about touch-range spells:
Most spells have ranges expressed in feet. Some spells can target only a creature (including you) that you touch. Other spells, such as the shield spell, affect only you. These spells have a range of self.
That's all it says about touch-range spells there. The only relevant restrictions would be those in the description of a spell or feature itself, as there are no general rules for it. It can simply be understood that any form of touch is sufficient for a touch-range spell.
5e rules designer Jeremy Crawford unofficially confirms this interpretation on Twitter:
Does a paladin's Lay On Hands feature require 'a free hand' - or can they use it as long as they are close enough to touch?
Lay on Hands requires you to touch the target. The feature isn't concerned with how you execute the touch.
Good to know I can ignore the title of the [feature] in determining how it works.
You are welcome to play the feature's name literally. In which case, I recommend laying both of your hands on the target.
Most groups play a feature like Lay on Hands with the name in mind—laying on a hand. Yet the rules don't say what the touch entails
Because "touch" doesn't have a special definition in the game, we rely on the general English definition of the word:
To make physical contact with; to bring the hand, finger or other part of the body into contact with.
This definition does not require specifically using one's hand to do so. As a result, any physical contact with one's body is sufficient. (As always, the DM is free to house-rule otherwise.)