= Solution
First suppose $\Delta$ has full dimension in $M_{\mathbb R}$. Form the <toric cone> over it in $M_{\mathbb R}\oplus\mathbb R$ and the graded <semigroup algebra>
$$
R_\Delta=\bigoplus_{l\ge0}\ \bigoplus_{m\in l\Delta\cap M} k\,\chi^m t^l,\qquad \deg(\chi^mt^l)=l,
$$
where degree zero contains only $k$. The lattice semigroup is finitely generated by the <Gordan lemma>. The <projective toric variety of a lattice polytope> is $\mathbb P_\Delta=\operatorname{Proj}R_\Delta$. Using all lattice points in all dilates avoids any assumption that the degree-one lattice points generate the ring.
For each vertex $v$ of $\Delta$, the degree-zero part after inverting $\chi^vt$ is
$$
(R_\Delta[(\chi^vt)^{-1}])_0=k[\operatorname{cone}(\Delta-v)\cap M].
$$
Indeed its exponents are $m-lv$ with $m\in l\Delta$. Conversely, every lattice point of the tangent <toric cone> has this form for sufficiently large integral $l$. These charts cover the <Proj construction>: each point of a dilate is a rational convex combination of vertices, so some power of its <monomial> is a product of vertex <monomials>. Their <dual cones> are the vertex <toric cones> of the inward <normal fan of a polytope>. The normal <toric fan> is complete because every <linear functional> on $\Delta$ attains its minimum on a <convex face>. This describes $\mathbb P_\Delta$ as a complete <toric variety>. Translating $\Delta$ by $m_0\in M$ gives the graded isomorphism $(m,l)\mapsto(m+lm_0,l)$, so containing the origin is unnecessary. If $\Delta$ is not full-dimensional, translate a lattice vertex to zero and use $M\cap\operatorname{span}_{\mathbb R}(\Delta-\Delta)$; the construction is then complete of dimension $\dim\Delta$.
For the simplex in question put $Q=\prod_iq_i$. Its inequalities are $x_i\ge0$ and $\sum_iq_ix_i\le Q$, so its inward normal <toric rays> are $\mathbb R_+e_i$ and $\mathbb R_+(-q_1,\ldots,-q_n)$. Here the $e_i$ in the dual space denote the standard dual basis. Apply the real linear map $A(e_i)=e_i/q_i$. It carries these <toric rays> and their <convex face> incidences to the standard <simplex fan>, with <toric rays> $e_1,\ldots,e_n,-\sum_i e_i$. The transported lattice is
$$
\boxed{N'=\bigoplus_{i=1}^n\frac1{q_i}\mathbb Z e_i\supseteq\mathbb Z^n,\qquad\mathbb P_\Delta\cong X_{\Sigma,N'}.}
$$
With $\gcd(q_1,\ldots,q_n)=1$, the primitive vectors of this <toric fan> in $N'$ are $v_i=e_i/q_i$ and $v_0=-\sum_i e_i$. The latter is primitive because $\lambda\sum_i e_i\in N'$ requires each $\lambda q_i$ to be integral, whose smallest positive solution is $\lambda=1$. Their unique primitive relation is
$$
v_0+q_1v_1+\cdots+q_nv_n=0.
$$
Thus mutual coprimality beyond the displayed common gcd condition is not needed.
For a general complete <toric variety>, introduce one variable $z_\rho$ for each <toric ray>. Its <toric homogeneous coordinate ring>, or <Cox ring>, is
$$
\boxed{S=k[z_\rho:\rho\in\Sigma(1)],\qquad\deg z_\rho=[F_\rho]\in\operatorname{Cl}(X_\Sigma).}
$$
The <toric divisor class sequence> is $0\to M\to\mathbb Z^{\Sigma(1)}\to\operatorname{Cl}(X_\Sigma)\to0$, where the first map is $m\mapsto(\langle m,v_\rho\rangle)_\rho$. Thus the graded piece $S_\alpha$ has basis the <monomials> $\prod z_\rho^{b_\rho}$ with $b_\rho\ge0$ and $\sum b_\rho[F_\rho]=\alpha$. For an invariant divisor $D=\sum a_\rho F_\rho$, <monomials> in degree $[D]$ correspond to <algebraic torus characters> $\chi^m$ through $b_\rho=a_\rho+\langle m,v_\rho\rangle\ge0$. This also identifies that graded piece with the global sections of $\mathcal O(D)$.
In our example $M'=(N')^*=\bigoplus_iq_i\mathbb Z e_i^*$, and its map to $\mathbb Z^{n+1}$ sends $\sum_iq_ib_ie_i^*$ to $(-\sum_iq_ib_i,b_1,\ldots,b_n)$. The quotient is therefore $\mathbb Z$, via $(c_0,\ldots,c_n)\mapsto c_0+\sum_iq_ic_i$. In particular $\deg z_0=1$ and $\deg z_i=q_i$, giving the explicit graded isomorphism
$$
\boxed{S\xrightarrow{\ \sim\ }k[Y_0,Y_1^{q_1},\ldots,Y_n^{q_n}],\qquad z_0\mapsto Y_0,\quad z_i\mapsto Y_i^{q_i}.}
$$
The target has the grading inherited from ordinary total degree in $k[Y_0,\ldots,Y_n]$. Its displayed generators are algebraically independent, which proves injectivity as well as surjectivity onto that subring.
To obtain the quotient, the <toric irrelevant ideal> is generated by $\prod_{\rho\notin\sigma(1)}z_\rho$. Here each maximal <toric cone> omits exactly one <toric ray>, so it is $(z_0,\ldots,z_n)$, and the open set is $\mathbb A^{n+1}\setminus\{0\}$. The <diagonalizable algebraic group> with <algebraic torus character> group $\operatorname{Cl}(X)=\mathbb Z$ is $k^*$. Its action on the Cox coordinates is dictated by their degrees, namely
$$
\boxed{\lambda\cdot(y_0,\ldots,y_n)=(\lambda y_0,\lambda^{q_1}y_1,\ldots,\lambda^{q_n}y_n),\qquad\mathbb P_\Delta\cong\mathbb P(1,q_1,\ldots,q_n).}
$$
Here the $y_i$ name the independent Cox coordinates; the graded-subring realization above does not change their weights. More concretely, the quotient chart where $y_i\ne0$ has coordinate ring $(S[y_i^{-1}])_0$, the invariant ring for this action. These charts glue to $\operatorname{Proj}S$ and to the specified <toric fan> charts. Positive weights make each orbit closed inside the punctured affine space, and in characteristic zero the finite stabilizers cause no failure of this <geometric quotient>. This is the <weighted projective space> construction.
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