The question should be posed in this way: given a real analytic $l$-form $\alpha$ on $M$, is there a real analytic form $l$-form $\widetilde \alpha$ defined on an open neighborhood of $M$ in $X$? The answer is yes. In fact there is a tubular neighborhood $U$ of $M$ in $X$ and a real analytic bundle map $p : U \to M$ such that $p_{|M} = \text{id}_M$ (this can be constructed by considering the exponential map of the normal bundle of $M$ in $X$ with respect to a real analytic metric on $X$, see Morrey). Then put $\widetilde \alpha = p^*(\alpha)$.

The question should be posed in this way: given a real analytic $l$-form $\alpha$ on $M$, is there a real analytic extension $\widetilde \alpha$ defined in an open neighborhood of $M$ in $X$? The answer is yes. In fact there is a tubular neighborhood $U$ of $M$ in $X$ and a real analytic bundle map $p : U \to M$ such that $p_{|M} = \text{id}_M$ (this can be constructed by considering the exponential map of the normal bundle of $M$ in $X$ with respect to a real analytic metric on $X$, see Morrey). Then put $\widetilde \alpha = p^*(\alpha)$.