I want to start a handyman business but I'd like to just keep a spreadsheet of all my transactions (including the name of customer, date, type of job, amount of check, any materials expenses) rather than making an invoice for each job because that's very time consuming. Is this legit?

Yes, this is perfectly fine. Keeping a spreadsheet is much better than having nothing to base your tax return on. The more records you have the better experience you'll have come tax time. Hopefully this helped!