It suffices to show this for l.o. free monoids on *finite* alphabets, as follows from the Compactness Theorem in logic - cf. any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid on the alphabet $X$, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

 1. the usual axioms for linearly ordered groups, expressed using
    $\circ$ and $\preceq$
 2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure
    that $M$ naturally injects as a subset of any model of $T$ via the
    interpretation of the $x_m$ constants]
 3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this
    injection an embedding of l.o. sets]
 4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the
    embedding a monoid homomorphism].

 
So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide a desired l.o. group, with $m\mapsto g_m$ (being the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of the axioms of $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m\in M$ so involved can be expressed as words in a finite subalphabet $Y\subseteq X$. This $Y$ generates a l.o. free submonoid $\mathbb S$ of $\mathbb M$ that contains all the $m\in M$ for which $x_m$ figures in statements occurring in $\Delta$. And so any l.o. group that "extends" $\mathbb S$ (in the sense of the OP) will be a model of $\Delta$.

Hence it is enough to consider free l.o. monoids on finite alphabets.