I know that many Orthodox Jews oppose abortion - presumably because they see this as a religious requirement - but does Orthodox Judaism see it as required or preferable to support a secular law criminalizing this practice? Are there any differences between modern Orthodox Judaism and Chareidi Judaism with respect to this topic?
Although Judaism prohibits abortion, including for non-Jews it doesn't necessarily prohibit all abortion. Certainly in the context of saving the life of the mother, abortion isn't only permitted, it is required.
That being the case, a question of a secular law which permits, but doesn't require or even sanction, prohibited behavior is much more of a political question, which will be heavily dependent on the political circumstances surrounding the question, potential adverse effects of the law, and how to approach it.
There is no single opinion on the matter, and although you could probably make a general statement that a Modern Orthodox Posek is more likely to be neutral to sympathetic to a legal situation permissive abortion than a Chareidi one, it cannot be said that there is a specific, clear, religious doctrine one way or the other, and I'm sure you will find exceptions in both directions to that general statement.
That is not to say that Judaism has nothing to say about the question. There are principles of Judaism at play here, besides the issue of abortion, there is also the issue of encouraging non-Jews to keep the Seven Noachide Laws. But any exact application is a specific Rabbinic ruling given the totality of the facts rather than any general Halachic statements one way or the other.
As noted in my answer to this question, abortion is generally prohibited, but is permitted in certain cases. All opinions agree that one is permitted - perhaps even obligated - to abort if the mother's life is in direct danger due to the pregnancy - perhaps even up to actual birthing. If the mother is not in direct danger, but will have negatively impacted health, most opinions hold that abortion is prohibited, but there are opinions which find ways in which to permit it.