While this is not the best place to ask this question, I will still try to give complete answer:
For a shorthand answer: No, unless you download it on your own.
You cannot hotlink this image in html document without letting execution of external scripts (which in this case would produce third party cookies).
From what I know, there is no cross-browser mechanism (or actually there is probably no such mechanism at all) that could allow download it client-side, while it has Access-Control-Allow-Origin property set not to allow cross-downloading of resources, and such it is in case of stackexchange server policy.
You CAN however manually download the file and store it on your server under specific filename and then use it as a regular image.
You can use various server-side solutions to download it at some cirumstances e.g. CRON, php script downloading the image at given conditions (e.g. periodically check if flair has changed, and overwrite it's copy if so), etc.
Here things get a little tricky.
While this flair is an image in technical terms, I believe this does not fall under art copyright, but most legislations are unclear on this all over the world. There are really many logotypes in this flair, it also contains user photo which basically makes it really messy in terms of being used by user it is attached to. I believe court would never find a person using this kind of flair guilty, but one cannot be certain of this. Note that there is no specified policy on flairs in stackexchange legal documents.
Hotlinking requires you to put cookie consent box within EU, as it WILL produce third party cookies.
What comes to downloading it as data client-side, if you were able to do it from your script client-side while all mechanisms concerning this DENY doing so,
this could really put you in risky situation, as this script would be most probably considered equally as computer virus in terms of law, so making a script like this, or even putting it on your site while you know how it works could in worst case get you to jail.
Downloading it manually on your own or by server-side script seems to be the safest way in terms of doing so. If only you are allowed to use it, there should be no problem with this.