You can search through cards in AnkiWeb by using the Search function. However, the things you can do in AnkiWeb are very limited, as you can not even delete cards IIRC. If you are serious about Anki, buying the AnkiMobile app is well worth it as donation and payment to keep the AnkiWeb servers up and running.
AnkiMobile supports searching through your cards with the same search parameters as in Anki for Desktop. Obviously, doing some things on mobile will be more tedious, but basically AnkiMobile (and AnkiDroid for Android) support the same things you could do on Desktop, with the exception of using add-ons.