Let $$(X_1,...,X_n)\sim F(x_1,...,x_n)$$ (not independent).
How can I sample from this distribution?
In the univariate case, on can use $F^{-1}(u),u\sim U(0,1)$. However, in the multivariate case these uniforms are not independent, such that they cannot be drawn independently.
Okay I think this will solve your problem and here less technical and probably what you want.
Added: Yes it is possible. So the main idea is to use the Bayes rule. Since with have the joint distribution, we can obtain the conditional marginals. For example for three events $A,B,C$:
$$P(A,B,C)=P(A|B,C)P(B,C)=P(A|B,C)P(B|C)P(C)$$ Now applying the same idea to random variables $X_1,X_2,X_2$, then $P(X_1<x_1,X_2<x_2,X_3<x_3)=...$ in this case we have all marginals but they are conditional. So they are basically all univariate and we can use for each of them the regular way $F^{-1}_i(U)$, $i=1,...$. Then what remains is to combine every sample and obtain the vector valued sample.