Once one observes the realisation $$X_1 = x_1, ..., X_o=x_p$$ the values $x_1,\ldots,x_o$ are fixed and known. Hence the conditional cdf (I assume this is a cdf) $$F(y_1, y_2 |X_1 = x_1, ..., X_p=x_p)$$ is a function of $(y_1,y_2)$ only and a regular cdf, for which standard simulation techniques apply.

Once one observes the realisation $$X_1 = x_1, ..., X_o=x_o$$ the values $x_1,\ldots,x_o$ are fixed and known. Hence the conditional cdf (I assume this is a cdf) $$F(y_1, y_2 |X_1 = x_1, ..., X_o=x_o)$$ is a function of $(y_1,y_2)$ only and a regular cdf, for which standard simulation techniques apply.