If you infringe upon a copyright, then you are responsible, not (only) your client/customer. You are creating the artwork and it’s your responsibility to act without negligence. Using artwork from a source you know is infringement is clear negligence. Beyond that, you’d need to speak to an attorney. And I am not an attorney. None of this should be seen as legal advice.
No one here can predict whether or not you’ll be sued. In the United States, anyone can sue anyone else, for anything. Whether or not they win is up to the judge. I see that site appears to be European-based. So, I don't know if things are different in Europe or your country of origin.
I would imagine if the client has a problem, they will contact the website, which will either provide your information to the client or contact you directly about the matter. Although you are essentially a third party, there's no protection merely because there is a middleman (website) between you and the end client.
Item 4 of that site's Terms of Service clearly reads:
(II) Postings. Users are responsible for all contents placed/posted on our site and resulting consequences.
And Item 8 reads:
In breach of copyright, trademark or other rights within or outside of a contest, it is up to the injured party to take legal action against the infringer and, where appropriate, to contact the relevant authorities for the protection of intellectual property. Brandsupply is not responsible for this.
Although Item 8 seems to infer more what to do if a client feels they have been infringed upon, it would seem to indicate an overall mindset that the client needs to take action against the infringing party, even it it is the user/uploader.
If you are concerned about this matter moving forward, you may consider being proactive and reaching out the website to inform them that you mistakenly used artwork in the logo which you do not have rights to use.