You could write to - General Register Office, GPO Building, Robb Street, Georgetown, Guyana, although I think you would need to know the year of birth, or at least a 5-10 year period to be searched. They hold the records from the 1880's to now. Like everywhere else, they would most probably charge a fee.
You could take a look at
www.candoo.com, a website re Caribbean genealogy, with a section on Guyana. You may find something more helpful there. Good luck.
Diana