Chinese Democracy was released on Sunday across much of the world
Axl Rose has accused Dr Pepper of failing to honour a pledge to give a free drink to all Americans if Guns N' Roses released a new album in 2008.
The drinks company made the promise after a number of release dates for the band's Chinese Democracy album - finally released this week - passed.
A lawyer for singer Rose said Dr Pepper's website crashed during the 24 hours it offered vouchers on Sunday.
The firm said it extended the scheme by 18 hours and had set up phone lines.
The voucher giveaway was timed to coincide with Sunday's US release of Chinese Democracy.
In a letter to Dr Pepper, Rose's lawyer Alan Gutman wrote: "The redemption scheme your company clumsily implemented for this offer was an unmitigated disaster which defrauded consumers and, in the eyes of vocal fans, ruined the day of Chinese Democracy's release."
Mr Gutman has asked the company for "an appropriate payment to our clients for the unauthorised use and abuse of their publicity and intellectual property rights".
And he has also demanded a full-page newspaper apology - to appear in The Wall Street Journal, USA Today, The New York Times and the Los Angeles Times.
A Dr Pepper spokesman did not respond directly to the requests but said it was "one of the largest responses we have ever received for a giveaway".
He said the company had gone to great lengths to fulfil requests including extending the promotion to 1800 EST on Monday and setting up a free phone line to accept voucher requests.
Earlier this week, China dismissed Chinese Democracy as a "venomous attack" on the nation.
An article in the Global Times, published by the ruling Communist party, said the album "turns its spear point on China".