Learn Spanish (Latin America): Ellos siguen apoyando mi carrera.

In Spanish, to say that someone "keeps doing" or "continues doing" an action, you use the verb seguir (to follow or continue) plus the gerund (the -ando or -iendo form of a verb). Here, siguen means "they continue" and apoyando is "supporting."

Why is the verb spelled siguen instead of "seguen"?

The verb seguir has a stem change in the present tense. The e changes to an i in all forms except for nosotros and vosotros. Because this sentence is about "them" (ellos), the verb becomes siguen.

Could I use the word soportando instead of apoyando here?

No, because that is a classic "false friend"! While apoyar means to support, back up, or encourage someone's career or goals, soportar means to tolerate or "put up with" someone or something.

Does carrera just mean "career"?

It means "career" in this context, but carrera is a very versatile word. Depending on the situation, it can also mean a "race" (like in running or cars) or a university "degree."