Yes, triggers are atomic in that they are executed in the same transaction as their parent query, and they either commit or roll back together. The parent query's transaction could be explicit, as suggested in Tab Alleman's answer, but that's unnecessary for a single statement since if you don't explicitly wrap the query in a transaction, you will get an implicit one, as discussed in the answers to this question.
Ok, but let's back up for a minute. You used the term atomic, but from your description, it seems that your actual goal is isolation rather than atomicity. It's a confusing subject because atomicity means two completely different things in the worlds of concurrent programming and databases:
In programming, atomicity pertains to mutual exclusion. A series of actions is atomic if it is uninterruptible by other processes. In the database world, that concept is instead referred to as isolation. Adding further confusion, atomicity is typically a binary concept, whereas database transaction isolation exists on a spectrum.
With databases, atomicity pertains to whether a partial failure is allowed among a series of actions. If a transaction is atomic, then either the entire sequence succeeds or none of it does. In-between states in which some actions succeeded and some failed are not permitted. This has nothing to do with excluding other users from resources while the sequence of actions is taking place.
In this case, the transaction might be adequate (such as if it is set to the strictest isolation level of "serializable"). But I'm no expert on isolation, and I'm not sure of the details of your situation. Various types of locks are another option.
More info:
- Another question in which someone was asking for atomicity when they really needed isolation. Apparently, using a lock was recommended in that particular situation.
- A great article (actually the first of a whole series) explaining the myths and confusion surrounding atomicity and the other characteristics of databases making up the acronym ACID
- SQL Server Transaction Isolation Levels