For your question to make sense, you have to actually have an infinite sequence $X_1,X_2,\dots$ of iid random vectors. Then, by the strong law of large numbers (SLLN) , $$\frac1n\,\sum_{i=1}^n f(X_i)\to Ef(X_1)=\int_{\mathbb R^p} f\, dp$$ almost surely (a.s.), which, as desired, implies the weak law of large numbers (WLLN) , that is, the convergence in probability (which is always implied by the a.s. convergence).
Concerning your second question, it is of course not true that "when a sequence of random variables converge[s] to a [constant] in probability, the convergence is a.s." -- That would be the same as to say that the convergence in probability always implies the a.s. convergence, because $Y_n\to Y$ a.s. (respectively, in probability) if and only if $Y_n-Y$ converges to the constant $0$ a.s. (respectively, in probability).