how can you tell if a travel agency is certified or legally operating?Is it polite to ask them if they are?

Question

is there a website where I can cross-check their credentials?

Answer 1

Yes you can ask them on which certification scheme they work, and you can check that.
Enter the name of the scheme in your search bar and if you get too many results add something like check, agent check or your chosen agents name.
If you go to the homepage of the organization there should be contact information too. So you can phone or e-mail to check.

Most travel agents have a big logo of the schemes they work with on or near the door and on their paper too, often with a number which should be right.
And as you often have to pay a little extra because they are in a scheme they should tell you, both in their selling talk and on the invoice.

If you can not find info about it, you can contact a national travel organization, (like the automobile club or consumer quality checking club) for if the people are legal.

As I do not know where you are from, and I am from a different country most likely, I can not tell you which schemes work where you live.

You can also have a look at the doors of other travel agencies in the area and see what schemes they work with, and check those websites for more info.