Which are considered correct?
1 I must login to / into / on the website.
2 I must go into / onto / on the website.
You can use (2) -- all three variations are OK -- but (1) is wrong. "Log in" is two words, though it is commonly spelled as (1) by people who don't know better.
You can tell from the -ing form that "log in" should be spelled as two words. We speak of "logging in," not of "logining."