The creditor can sue you but whether you are liable will depend upon how your business was structured and whether you personally guaranteed payment to the web developer. Hopefully you had an attorney when you started your LLC. Return to the attorney for a quick consultation or consult with a new attorney who handles business organizations. You might be personally protected, and further might be able to demand the creditor cease and desist under the Fair Debt Collections Practices Act. All will depend upon the nature of your agreement with the developer and the manner of organization of the defunct business.
You are asking a number of questions, actually. Here they are:
(1) Can you be sued for obligations of a company that was dissolved or ceased to exist?
(2) If yes, can you be served with process where you work; and what are your options?
First of all, if your LLC has truly been administratively dissolved (don't assume, make sure) and you hired a web developer on behalf of the LLC which acting as Managing Member and owner of the LLC then the "limited liability" benefit of the LLC ceases to protect you. The idea is that if anyone could dissolve their LLC without settling up with creditors then there would be chaos. But if your LLC is still in force and you acted on behalf of the company only, then you may yet escape personal liability.
Second, regardless of the answer above, you can be served at work. It's as simple as that. Service may be had in any location where you can be found, including your place of work.
But that's not the end of the analysis, because being served at work seldom results in job loss. Why would it? People have lives, get into disputes, and sue one another. Unless I am missing something, simply being served by a Sheriff is no reason to lose your employment. But since you are concerned about this issue you should definitely speak with a knowledgeable lawyer who can resolve the matter - whether or not that involves going to Court.
I hope this information has been helpful. Feel free to contact my office for more information.