If I understand the question, the short answer is "yes, you can freely and functorially adjoin infinite products to monoids". The basic idea is that algebraic theories can accommodate arbitrary arities (bounded above by some cardinal), and one can discuss relative free-forgetful adjunctions between categories of algebras in great generality.

At first pass, let me just focus on the purely monoid-like aspects for now, because those are easier to visualize. Once the story for that is clear, one can work in inverses.

As a warm-up, let's recall that one way of defining an ordinary monoid $M$ is as an algebra of the terminal nonpermutative operad. In plainer English, this means we a single operation

$$\mu_n: M^n \to M$$

for each finite ordinal $n$, so that

$$\mu_{n_1 + \ldots + n_k} = \mu_k(\mu_{n_1} \times \ldots \times \mu_{n_k})$$

(generalized associativity equation).

Now let's generalize operads so as to allow operations of countably infinite arity. By "arities", I will really mean countable ordinals. Given an ordinal $k \lt \aleph_1$ and ordinals $n_j \lt \aleph_1$ for $j \lt k$, you can concatenate the $n_j$ to get a new ordinal $\sum_j n_j \lt \aleph_1$. Concatenation is associative in an evident sense. Now define an $\omega$-monoid to be a set $M$ equipped with operations

$$\mu_k: \hom(k, M) \to M,$$

one for each $k \lt \aleph_1$, such that $\mu_{\sum_j n_j} = \mu_k (\prod_j \mu_{n_j})$. (I am not completely certain we have to go all the way up to $\aleph_1$, but if not it will be some suitable initial segment. Let's just say $\aleph_1$ for now.) This condition can be interpreted in any category with countable products, such as $Set$.

The free $\omega$-monoid on a set $X$ will be $\sum_{k \lt \aleph_1} \hom(k, X)$. We get in this way a monad $T$ for a free-forgetful adjunction between $\omega$-monoids and sets.

There is a general bit of nonsense that for any morphism of monads $\phi: S \to T$ on $Set$, there is a forgetful functor $Alg_T \to Alg_S$, and this forgetful functor has a left adjoint. This follows from an adjoint functor theorem, although if I'm not mistaken, in this particular scenario a more direct construction is available: if $S$ is the monad for ordinary monoids and $T$ is as above, the evident inclusion $S \to T$ induces a forgetful functor

$$\omega-Mon \to Mon$$

which has a left adjoint $L$ described by the type of coequalizer familiar from tensor products:

$$L(M) = coeq((\mu_T \circ \phi) M, T\theta: TSM \stackrel{\to}{\to} T M)$$

where $\mu_T: TT \to T$ is the monad multiplication and $\theta: S M \to M$ is the structure of $M$ as $S$-algebra. This left adjoint $L$ would correspond to what I think you were asking for with "freely adjoined colimits", and the left adjoint means we indeed have a functorial construction.

If you want to work inverses in, you can do that too. Long story short: for any set of formal operation symbols of arbitrary arity, subject to any set of well-formed equations you jolly well please, you can form a monad whose algebras are precisely the models of for the corresponding algebraic theory. So: together with operations of countable arity as above, subject to generalized associativity equations, you can certainly toss in an unary inversion operation as well. I leave it to you to decide what, in addition to associativity, are the sensible equations to impose on inversion $i$, but it seems to me you might want to impose only

$$\mu_2(id \times i) = id = \mu_2(i \times id)$$

and stop there. (Operations involving infinitely many instances of inversion are still permissible, but the equations would enforce only finitely many cancellations at a time.) You get in this way a monad for "$\omega$-groups", and again the forgetful functor from $\omega$-groups to groups admits a left adjoint, constructed in a way analogous to the above.