French special letter "oe" is not an option
I don't have this character on my MS French keyboard; AND I don't see it on DL screen: I am marked wrong every time :
Under the Duolingo text entry area, I have about 12 characters for French. œ is the last of them. In capitals it looks like a reversed D followed by an E, rather than an O followed by an E.
I use Firefox mobile (in desktop view), which is not Duolingo's favoured brower (for example, audio cuts off 1 second or so before the clip finishes). Nevertheless these letters show.
If it's really not there, I agree with Qiunn's advice - copy and paste it. You can copy it from here œ or from Captain_Duo's link. And (presuming you are not using a mobile view) if it is missing you might want to report it to Duolingo, and mention which browser you are using: http://support.duolingo.com/hc/en-us/requests/new
You can access all accents even without a French keyboard. You simply hold down the letter "o" for options ô ö ò ó œ ø ō õ. Same with every other letter that has accents.
This is for mobile, of course. It depends on your virtual keyboard, though. I am using Google's French, English and Turkish virtual keyboards, and I switch from one to the other when needed.
On a desktop or laptop, it is also possible to make your keyboard act like a French keyboard even if it is an English one, thanks to the language bar. You will have to know where the keys are on a French keyboard, though.
However, Œ and œ are special letters, as they were left out of the Latin 1 set of characters (ISO 8859-1), and even worse: they are absent from all the French (physical) keyboards. They can be typed with ALT + 0140 and ALT + 0156, though. Not easy.
In the future, all keyboards will probably be customisable. Each key will display the character currently assigned to it. It will be easy to switch from an English keyboard to a French, Russian, or Chinese keyboard, and we will be able to have our own customised version of these. I would gladly replace µ and ¤, that I do not use, with œ and Œ, that I regularly need. Well, it may already be possible to do that on a normal keyboard...
And yes, it is. With the Microsoft Keyboard Layout Creator, if you are using Windows. To download it: https://www.microsoft.com/en-us/download/details.aspx?id=22339. Tutorial: https://www.pcmag.com/news/353425/how-to-customize-your-computer-keyboard.
I've never downloaded any French keyboard, and I can still do it with the desktop version. I have a mac though. Not sure if they have that already installed.