Can my travel agency software show my booking site and my customer emails in another language?

Yes, up to a point. The parts of the system that belong to you, such as the labels on your booking site, your email templates and the wording on your quotes and vouchers, can normally be set up in another language, and a customer can be shown the version that suits them. Anything you write yourself, like a tour description or a welcome note, you would supply in each language you want to offer.

Where it becomes less straightforward is the content that comes from your suppliers. Hotel descriptions, room names, fare conditions and cancellation rules arrive in whatever language the supplier sends, and that is very often English only. Nothing in the middle can reliably rewrite those into another language without risking a mistake in a rule your customer is relying on, so you will usually see supplier wording in its original form even on a translated page.

How much of this you get also depends on your plan and on how your site was set up, and adding a new language later is a piece of work rather than a switch, because every template and every piece of your own text has to be written again. Dates, names and address formats may also need attention so they read naturally to a local reader.

A good next step is to write down the languages your customers actually ask for, look at one of your own emails and one hotel result side by side, and ask your provider what is covered for those languages and what you would need to supply yourself.