Maybe overkill, but elegant:

By a theorem of Hirsch, an oriented $3$-manifold $M$ embeds in the $5$-sphere (nonorientable case: Rochlin and Wall, independently). By Alexander duality, $M$ bounds a "Seifert $4$-manifold."

Maybe overkill, but elegant:

By a theorem of Hirsch, an oriented $3$-manifold $M$ embeds in the $5$-sphere (nonorientable case: Rohlin and Wall, independently). By Alexander duality, $M$ bounds a "Seifert $4$-manifold."

(Some references:

Hirsch, Immersions of almost parallelizable manifolds. Proc. Amer. Math. Soc. 12 1961 845–846.

Rohlin, The embedding of non-orientable three-manifolds into five-dimensional Euclidean space. Dokl. Akad. Nauk SSSR 160 1965 549–551.

Wall, All 3-manifolds imbed in 5-space. Bull. Amer. Math. Soc. 71 1965 564–567. )