Does hospitalo exist, or is malsanulejo the only word that Esperantists use?
Both of these words have been used since the very early years of Esperanto, with precisely the same meaning. In the meantime, many other similar pairs of words, one based on an international word and the other based on Esperanto work formation, have appeared (e.g. "egoismo" vs. "memismo", "ornitologio" vs. "birdoscienco", "kardiologo" vs. "korkuracisto", "glaĉero" vs. "glacirivero" or "glaciejo"), and in the case of some of them, one has managed to push the other out of usage (e.g. "komputilo" has pushed "komputoro" and "komputero" out of usage), but "malsanulejo" and "hospitalo" continue to strive in parallel for more than a century, without one managing to push the other one out of usage.
I personally prefer to use compound words in case of doubt, as many of the purportedly international words are not widely known to speakers of non-European languages, so avoiding them can decrease their disadvantage when learning Esperanto a little bit.
According to the vortaro.net, the word "hospitalo" exists : http://vortaro.net/#hospitalo
But I've never seen it before today :-)
It's not a good idea to use new roots because some non-european learners will have more difficulties to learn Esperanto and to understand you.