Both VAE and LDA (latent Dirichlet allocation) is based on variational inference, and they both try to optimize ELBO objective function
Variational autoencoders use reparameterization so that "backprop can be applied without being blocked by sampling."
In LDA (latent Dirichlet allocation), the model also sample a hidden variable (z) and use this z as parameter to generate other observed variable (word).
LDA can be solved by a nice/clear variational inference (coordinate ascent mean-field algorithm), without any "reparameterization trick". Then why "reparameterization trick" is a must-have for VAE?