Learn Arabic (Modern Standard): حجزنا الغرفة عن طريق الإنترنت.

The literal translation of عن طريق is "from the path of" or "by way of". In Arabic, this is the standard compound preposition used to express doing something "via", "through", or "by means of", such as booking a room through the internet.

Why doesn't the word طريق have the definite article ال (al-)?

The words طريق and الإنترنت form an idafa (a possessive or "of" construction), literally meaning "the path of the internet". In an idafa, the first word never takes ال, but the phrase as a whole is still definite because the final word (الإنترنت) is definite.

What is the root of حجزنا and what is its core meaning?

The verb حجزنا (we booked/reserved) comes from the root ح-ج-ز. The core meaning of this root is to hold back, block off, or reserve. You might also recognize this root in the noun حاجز which means a barrier, hurdle, or checkpoint.

Does the loanword الإنترنت follow normal Arabic case rules here?

Mostly, yes. Grammatically, it is in the genitive case here because it is the second part of an idafa. However, because it is a foreign loanword ending in a consonant, speakers usually do not pronounce a case ending on it in everyday speech, though in highly formal MSA it could take a Kasra (i) sound at the end.