Yes, there are several websites that can be used for this.
Probably the best-known would be leetchi.com.
This is a French company whose site is for setting up a cagnotte en-ligne – “online kitty”.
You can set up a cagnotte for many different things.
You start by naming your cagnotte and clicking to say what kind it is, from a projet solidaire (“for a good cause”), cadeau commun (group gift) or dépense à plusieurs (expense to be met by several people).
Within each category there are sub-headings and in this case it might go under médical perhaps, or autre projet solidaire.
You can then share on social media and start collecting.
There is no fee for setting up or collecting on this site but once you have finished there is a commission for sending the money by bank transfer, from 1.9% to 4%, depending on the amount collected.
You can also spend the money free on partner sites or pass on the cagnotte to the beneficiary to use as they want (ie. a transfer or spending on partner sites).
Colleo.fr is similar and takes a set 3.5% commission on every donation. There is no obligation to spend on partner sites and transferring money to your bank account is free.
There is also lepotsolidaire.fr, a page for collecting for good causes run by le pot commun, a site which was originally set up for reasons such as collecting for a group gift.
This takes 2.9%-4% off at the transfer stage.
Finally, GoFundMe is an American firm with a French branch at fr.gofundme.com.
It gives the impression of being free but the small print says there is a “small commission” of 2.9% + €0.25 covering costs of dealing with credit card payments and secure transfer of funds.