Learn Spanish (Spain): Ellos no encuentran la llave.

The verb here is encontrar (to find). It is a stem-changing verb. This means that when you conjugate it, the o in the stem changes to ue for most forms. Since ellos (they) is one of those forms, encontrar becomes encuentran.

Why is there no word for "do" in the Spanish sentence?

In English, we use helper words like "do" or "does" to make a negative sentence (like "they do not find"). Spanish is simpler! You just put no directly in front of the verb. So, no encuentran perfectly translates to "they do not find" or "they don't find".

The word llave ends in "-e". How do I know it uses la instead of el?

While words ending in -o are usually masculine and words ending in -a are usually feminine, words ending in -e can be either. You just have to memorize their gender when you learn the vocabulary. Llave (key) is feminine, so it takes the feminine article la.

Could I drop the word Ellos and just say No encuentran la llave?

Yes, absolutely! In Spanish, the ending of the verb (-an in encuentran) already tells us that the subject is "they" or "you all" plural. If it is already clear from the context who you are talking about, it is very common and natural to drop the pronoun Ellos.