Yes, you could have liability if the logo is infringing. You would want to have a comprehensive services agreement with a warranty that the logo is not infringing and that the business would indemnify you for any legal action against you or your company.
However, having such language is not a guarantee that you will not incur any liability or certainly legal fees. As such, if you feel there is a real risk of infringement, don't do it. Or at least consider the risk in the price of the services.
Ifyou handle this properly, YOU won't, but the website owner will. You should fall under the "innocent printers'" exception to the infringement act. 15 USC 1114(2)(A). http://www.law.cornell.edu/uscode/text/15/1114 Get a document from the website owner that they warrant they own the image and will hold you harmless from any liability for its use so you can credibly claim to be an innocent printer entitled to the protection under the Act and have documentation to prove it.
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You certainly could be liable for copyright infringement. At a minimum you should want a contract with them that deals with the issue, and also has them agreeing to indemnify you if you were sued. It is very possible that something simply downloaded from the internet would infringe on someone's copyright or other intellectual property rights.
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Yes, you could be liable for direct copyright infringement, contributory infringement, and trademark infringement (or contributory trademark infringement). Your changes to the logo create a derivative work--it is illegal for you to make such a derivative work without permission of the copyright owner.
Because of this potential liability, you should insist that this business enter into a written agreement that indemnifies you from liability in the event you are sued for infringement of copyright and trademark law. Web-site designers should always protect themselves by performing their services pursuant to written agreements.
My guess is that this issue regarding the logo is the tip of the iceberg here. There probably are also serious issues under patent law---who is going to obtain a clearance analysis to ascertain whether the web-site you design violates third party patent rights? Is that your responsibility? Most businesses require web-site designers to indemnify them in the event the designs infringe third party IP rights? Are you in a position to know whether the web-site you design here will infringe third party patent rights or other IP rights? Are you in a position to know whether the web-site operations proposed by this business will violate business method patents? If not, then you need legal help on these issues.
The web-site design business can be lucrative but it is also fraught with risks. You need to work closely with legal counsel to deal with a wide array of issues. You need to insist on written contracts with any business that wants to hire you to perform these services. And these contracts need to sort through a wide range of IP law issues. And most importantly, you want the company to take charge of obtaining IP law clearances that they need to safely operate the web-site, and to indemnify you in case there are violations of IP law. Of course, smart businesses require their web-site designers to indemnify them if the web-site designs violate copyright or patent law----and the big boys in the web-site design field are capable of providing such indemnification because they work closely with legal counsel to assure that the web-sites they create do not violate third party IP rights, and they obtain intellectual property insurance to protect them in the event they are sued.
All of this seems a little complicated---you bet it is. That is why you need a lawyer before taking on this project.