Yes, any smooth manifold can be embedded as a Lagrangian in a closed symplectic manifold. In fact, Lisca and Matic proved that any Stein domain (like $T^*L$) embeds into a smooth projective variety with ample canonical bundle, so you can do better than just any old closed symplectic manifold. Here's a citation:

- Lisca, P., Matić, G. Tight contact structures and Seiberg-Witten invariants. Invent. Math. 129 (1997), no.3, 509–525.

Going a little further, your question is in fact equivalent to asking whether the unit cotangent bundle admits a "symplectic cap", which can then be glued to $T^*L$ to give a closed symplectic manifold. (The not-as-obvious implication is given by the Weinstein tubular neighborhood theorem.) In fact, it is now known more generally that any contact manifold admits a symplectic cap. This was proved in three dimensions by Etnyre and Honda and in arbitrary dimensions independently by Lazarev as well as by Conway and Etnyre.

- Etnyre, J., Honda, K. On symplectic cobordisms. Math. Ann. 323 (2002), no.1, 31–39.
- Lazarev, O. Maximal contact and symplectic structures. J. Topol. 13 (2020), no.3, 1058–1083.
- Conway, J., Etnyre, J. Contact surgery and symplectic caps. Bull. Lond. Math. Soc. 52 (2020), no.2, 379–394.