I have a small claims judgment against a corporation (LLC). However, the business is now apparently operating under a different name, same owner. I was able to determine that both LLC names were registered in my state (Missouri) prior to the case. Calls to the phone number of the "first" business are now answered in the name of the "second" one.
If the judgment I have is against the "first" LLC's name, but there is no bank account in that name, only an account is in the "second" LLC's name, what would I need to do in order to request a garnishment order against the "second" LLC's bank account?

You need to get a new judgment against the owner, on the basis that, somehow, the limited liability afforded the owner should be ignored. In the case of corporations, this is referred to as "piercing the corporate veil".

BTW, corporations and LLCs are not the same thing. The LLC is not a corporation.

I almost missed your follow-up question because it was made in a "comment". :)

If you were to sue the "organizer", plus other "John Doe" parties (place-holder defendant names to be amended later), you could engage in discovery to find the evidence (see the link to "piercing the corporate veil" I gave you earlier). Then use that evidence to get the names of the other principals, if any, and amend your complaint. Serve each individual with process. Then proceed to get a joint-and-several judgment against all of them.

If you can get a judgment against one or more owners of each LLC, you should also be able to get a garnishment order against the bank accounts of each LLC.

All of the above will probably require a considerable amount of work, but that's how I would proceed if it appears to be worth the effort.

AK lawyer is, of course, exactly right in his response. Let me add that piercing the corporate veil is neither an easy nor an inexpensive task. It takes quite a lot of evidence in most courts with which I am familiar to persuade the judge to pierce the veil.