Learn Russian: Она сама пришла.

Вся means "all" or "the whole," while сама comes from the word сам, meaning "oneself" or "by oneself." If you said Она вся пришла, it would sound strange, like "Her entire body arrived." Here, Она сама пришла simply means "She came herself" or "She came on her own."

What would this sentence look like if a man came by himself?

You would change both the "oneself" word and the verb to their masculine forms: Он сам пришел. If it were a group of people, you would use the plural forms: Они сами пришли.

Why use the verb пришла instead of шла?

Пришла means "arrived" or "has come" (a completed action with a result). Шла just means "was walking" or "was on her way" (an ongoing action). Because she successfully arrived, we use the completed verb пришла.

Can I change the word order to Она пришла сама?

Yes, Russian word order is very flexible! Она сама пришла is a very natural way to say it. Putting сама at the very end (Она пришла сама) puts even more emphasis on the fact that she did it completely on her own, without anyone else's help.