Let $i\colon X\to Y$ be an embedding of two smooth and compact manifolds (without boundary) and let $N_iX$ be the normal bundle of this embedding. A Pontrjagin-Thom construction is a map $$ c_i\colon Y \to Th(N_i) $$ (which is only unique up to homotopy as one chooses a tubular neighborhood (see this ncatlab page for details)).
Is $c_i$ necessarily null-homotopic, if the normal bundle $N_i$ is trivial?
No, look at $S^1 \subset S^2$. The map is $S^2 \to S^2/(S^2-S^1\times (-1,1))\stackrel{\simeq}{\to} S^2\vee S^1$ which has non trivial degree (check the induced morphism on second cohomology).