It seems like there is an extra word, or possibly a missing word. "Have to their lawful rights" doesn't make any sense. "Have their lawful rights" makes sense. The writer could also have meant something like "have to use their lawful rights". What is currently written seems like an error.
Your suggestions "reach" or "get" would work, but colloquially I would choose "receive", or if they are fighting hard for these rights (instead of just receiving them), "achieve" or "attain".
Additionally, "rightful rights" is very redundant. That phrase means "the rights to which they have a right". I would just leave it as "lawful rights".