The actress - who portrayed the dim but lovable Alice Tinker opposite French's Geraldine Granger in the long-running BBC comedy - died of natural causes on Wednesday evening, her agent said.
Emma also starred as Hugh Grant's sister Honey in Richard Curtis' Bafta-winning film Notting Hill.
Ms French said: "Emma was a very bright spark and the most loyal & loving friend anyone could wish for.
"I will miss her very much."
Recalling their characters' close friendship, French shared a photo of Alice lying on top of Geraldine in a shot from the popular series.
She wrote on Twitter: "I was regularly humped like this by the unique & beautiful spark that was Emma Chambers. I never minded. I loved her. A lot."
Mr Grant called news of her death "very sad".
He wrote on Twitter: "Emma Chambers was a hilarious and very warm person and of course a brilliant actress.
The Vicar Of Dibley, also written by Curtis, originally ran from 1994 to 1998 but returned for numerous festive and comic relief specials with the latest episode airing as recently as 2015.
Each episode would close with Geraldine telling a joke to Alice who would either react nonplussed, interpret it literally, or only understand the humour once it was explained.
Curtis' wife, broadcaster Emma Freud, was among one of the first to pay tribute to the couple's "beautiful friend".
She wrote on Twitter: "We're very very sad. She was a great, great comedy performer, and a truly fine actress.
"And a tender, sweet, funny, unusual, loving human being."
Freud posted a clip of Chambers and French in one of the joke scenes from an episode in 2005, and wrote: "How could you not love this girl. Emma Chambers... thank you for the brilliance."
Former Top Gear presenter Jeremy Clarkson, who also grew up in Doncaster near Chambers, wrote: "I'm sad about Emma Chambers.
"Knew her when she was a kid in Doncaster. She was very funny."
Chambers' agent John Grant said in a statement: "We are very sad to announce the untimely death, from natural causes, of the acclaimed actress, Emma Chambers.
"Over the years, Emma created a wealth of characters and an immense body of work.
"She brought laughter and joy to many, and will be greatly missed.
"At this difficult time we ask that the privacy of the family and loved ones be respected."
Citizen Khan star Adil Ray tweeted: "So sad. Emma Chambers was absolutely masterful. Very funny, she made the daftness believable. The joke coda with @Dawn_French in Dibley required great skill and Emma just nailed it. She made it. We say this a lot, but Emma IS a great loss to our screens."
Jon Plowman, executive producer of The Vicar Of Dibley and former head of comedy at the BBC, said: "This is a sad day. Emma was a gifted comic actress who made any part she played - no matter how ditzy or other worldly - look easy.
"To create a much-loved comic character as she did, you have to be every bit as bright and clever as Emma always was.
"She was great fun to work with and adored by all the cast and crew of Vicar Of Dibley. She will be missed and our deepest condolences go out to her family and friends."