The word alright does exist, but it’s much more acceptable to use all
rightall right in American English. Both alright and all right are acceptable in British English.
1. The American Heritage Dictionary of the English Language (3/e, Houghton Mifflin, 1996) has this as its entry for alright:
"Non-Standard. All right." In its usage notes for all right, it says this:
All right, usually pronounced as if it were a single word, probably
should have followed the same orthographic development as already and altogether.
But despite its use by a number of reputable authors, the spelling alright has
never been accepted as a standard variant, and the writer who chooses to risk
that spelling had best be confident that readers will acknowledge it as a token
of willful unconventionality rather than as a mark of ignorance.
2. Michael
Swan, in Practical English Usage (2/e, Oxford University
Press, 1995), says:
The standard spelling is all right. Alright is
common, but many people consider it incorrect.
3. Brian A. Garner, in The
Oxford Dictionary of American Usage and Style (Oxford
University Press, 2000), comments:
All right. So spelled. The one-word spelling (alright) has
never been accepted as standard in AmE.
4. The New York Times Manual of Style
and Usage (Random House, 1999)
states
succinctly: "all right (never alright)."
5. Ann Raimes, in Keys for Writers (Houghton Mifflin, 1996), says: "Alright is
nonstandard. All right is
standard."
There is, though, a certain acceptance of alright, as it exists, descriptively,
in:
The Collins COBUILD English Dictionary (HarperCollins, 1995):
Under “alright?
alright. See all right
Under “all right?
all right; also spelled alright
The Collins Cobuild online
(https://titania.cobuild.collins.co.uk/form.html).
This corpus lists both "all
right" and "alright," almost all in conversational contexts. In examples
from American English, "alright" appeared in only 28 instances (out of a
possible 40 appearances), and "all right" in 40 (the maximum that ever appears
in these sample examples). In British English, "alright" occurred in the
maximum 40, and "all right" also
occurred in the maximum 40.
The Economist online (https://www.economist.com/search/search.cfm?cb=46&page=index&qr=%22all+right%22&area=1):
Both "alright" and "all right" occur in news articles.
So, with this information, it seems that you would always be correct using all right,
particularly in AmE. In BrE you can use alright, but all right is
equally acceptable.