(I have added a new section to this answer, with a list of references at the end.)
It's not easy to find clear and uncluttered statements of Hölder's theorem (1901) and its many generalisations, so I've cobbled together what I think are the definitions and theorems most relevant to the present question. Page references are to Fuchs [4].
The given definition of $\leq$ makes the ordered additive structure of $M$ a naturally ordered semigroup [p.154].
Axiom (3) states that the ordered additive structure of $M$ is a cancellative semigroup [also p.154].
By Lemma B on p.163, an Archimedean totally ordered cancellative semigroup has no "anomalous pairs". What these are does not matter here! What does matter is that by Theorem 4 (Alimov) on p.167, a totally ordered semigroup $S$ is o-isomorphic [p.21 - the same as what is meant in the question by an "embedding" (onto), except that multiplication need not be preserved] to a subsemigroup of the additive group of $\mathbb{R}$ if, and only if, it satisfies the conditions:
(a) $S$ contains no anomalous pairs,
(b) $S$ is cancellative.
Therefore, an Archimedean totally ordered cancellative semigroup is o-isomorphic to a subsemigroup of the additive group of $\mathbb{R}$.
Theorem 2 (Hölder, Clifford) on p.165 is followed by the remark that any two o-isomorphisms of $S$ into $\mathbb{R}^{\geqslant0}$ "differ merely in a positive real factor".
Greater emphasis is given to this proposition by Behrend [1] and Krantz et al. [6].
It implies that if $\xi$ is any non-negative real number, there is a unique order-preserving additive homomorphism [this clumsy form of words is chosen to allow the possibility that $\xi = 0$], $f_\xi: M \to \mathbb{R}^{\geqslant0}$, such that $f_\xi(1) = \xi$.
(We can take $\mathbb{R}^{\geqslant0}$, rather than $\mathbb{R}$, as the codomain, because for all $x \in M$, $x \geq 0$, and therefore $f_\xi(x) \geq 0$.)
For each $x \in M$, the order-preserving additive homomorphisms $y \mapsto f_1(xy)$ and $y \mapsto f_1(x)f_1(y)$ both satisfy $1 \mapsto f_1(x)$. By the uniqueness clause in the result above, the two functions are equal, i.e., $f_1(xy) = f_1(x)f_1(y)$ for all $x, y \in M$. Therefore, $f_1: M \to \mathbb{R}^{\geqslant0}$ is an embedding in the sense required. $\square$
(It is noted in the following section, however, that statement (6) of the Archimedean axiom is not strong enough.)
The point of closest approach in the literature (surveyed
haphazardly, and without access to MathSciNet) seems to be section
2.2.7, "Archimedean Ordered Semirings", especially Theorem 6,
on pp.54-58 of the book by Krantz et al. [6].
Krantz et al. define an "ordered local semiring", in which
addition and multiplication may not be defined for all pairs of
values of their arguments. Dispensing with this generality, for the
purposes of this answer only, call a quadruple $\left\langle A,
\geqslant, +, \cdot \right\rangle$ an ordered hemidemisemiring
if $A$ is totally ordered by $\geqslant$, $+$ and $\cdot$ are
associative binary operations on $A$, $\cdot$ distributes over $+$
to the left and to the right, and if $a \geqslant b$, then
$a + c \geqslant b + c$, $c + a \geqslant c + b$, $ac \geqslant bc$,
and $ca \geqslant cb$.
(Note: neither $+$ nor $\cdot$ is presupposed to be commutative.)
$\left\langle A, \geqslant, +, \cdot \right\rangle$ is called
positive iff $a + b > a$ for all $a, b \in A$.
A positive hemidemisemiring $\left\langle A, \geqslant, +, \cdot
\right\rangle$ is called regular iff, for all $a, b \in A$
such that $a > b$, there exists $c \in A$ such that $a \geqslant b + c$.
$\left\langle A, \geqslant, +, \cdot \right\rangle$ is called
Archimedean iff, for all $a, b \in A$, there exists $n \in
\mathbb{N}$ such that $na \geqslant b$.
After specialisation and transcription into the new notation and
terminology, Theorem 6 in Krantz et al. becomes:
Let $\left\langle A, \geqslant, +, \cdot \right\rangle$ be an
Archimedean, regular, positive ordered hemidemisemiring. Then
there is a unique function $\phi: A \to \mathbb{R}^{>0}$ such that,
for all $a, b \in A$,
(i) $a \geqslant b$ if and only if $\phi(a) \geqslant \phi(b)$;
(ii) $\phi(a + b) = \phi(a) + \phi(b)$;
(iii) $\phi(ab) = \phi(a)\phi(b)$.
Given a PM-semiring, $M$, let $A = M \setminus\{0\}$. By (2) and
(5), the addition and multiplication operations on $M$ restrict to
operations on $A$, which also inherits the total ordering of $M$.
It follows from the postulated properties of $M$, together with the
definition of $\geqslant$, that $A$ is a regular, positive ordered
hemidemisemiring.
Here we must pause to note that postulate (6) requires modification,
because it fails to rule out the existence of infinitesimal elements,
which prevent $M$ from being embedded in $\mathbb{R}^{\geqslant0}$.
For a counterexample, take $M = A \cup \{0\}$, where:
$$
A = \{ a_dt^d + a_{d+1}t^{d+1} + a_{d+2}t^{d+2} + \cdots
\in \mathbb{Z}[t] \mid d \in \mathbb{N}_0, \ a_d > 0 \}.
$$
This is the positive cone of a total ordering of $\mathbb{Z}[t]$,
which is compatible with the ring structure. Postulates (1) to (6)
are all satisfied, i.e. $M$ is a PM-semiring, but the element
$t \in M$ is a positive infinitesimal, i.e. $t > 0$ and $nt < 1$
for all $n \in \mathbb{N}$, therefore there can be no embedding
$M \to \mathbb{R}^{\geqslant0}$.
Supposing provisionally that (6) is replaced with the usual
Archimedean axiom, Theorem 6 of Krantz et al. applies.
Let $\phi: A \to \mathbb{R}^{>0}$ be the function supplied by the
theorem. If we extend it to $M$ by defining $\phi(0) = 0$, it is
clear that $\phi: M \to \mathbb{R}^{\geqslant0}$ is an embedding.
$\square$
References (not limited to those actually used here, but with no pretension to completeness):
[1] Felix A. Behrend, A contribution to the theory of magnitudes and the foundations of analysis (1956)
[2] Damon Binder, Non-Anomalous Semigroups and Real Numbers (2016)
[3] A. H. Clifford, Totally ordered commutative semigroups (1958)
[4] László Fuchs, Partially Ordered Algebraic Systems (Pergamon 1963, repr. Dover 2011)
[5] E. V. Huntington, A complete set of postulates for the theory of absolute
continuous magnitude (1902)
[6] David H. Krantz et al., Foundations of Measurement, I: Additive and Polynomial Representations (Academic Press 1971, repr. Dover 2007)
[7] Dana Scott, A General Theory of Magnitudes (unpublished, but referred to in this answer) (1963)
[8] Hassler Whitney, The Mathematics of Physical Quantities (1968)
