NO - you cannot access the COM port of the client from the web
Browsers should not expose the hardware to the web. You really do not want to be able to access a web page and have it be able to query your ports and use them to send or receive signals.
While some hardware options has been abstracted and made available to the web (such as microphones and cameras), this is not the case of COM ports.
This is the way it should be. Therefore, if you need a solution to work only with web technologies you are out of luck.
You could access the COM port of the server from the web
If you need to expose the COM port of a computer to the web such that you can use it remotely, what you could do is set up a server on the computer and use server side code to access the COM port.
Doing this is not advice. If you want to do this, you will have to not only worry of having a authentication to prevent sure random people access your hardware, but also to prevent multiple clients using the same port at the same time.
There is code available in the common server side languages (such as Java, Python, even PHP) that you could be adapted for this purpose.
You could hack your way to the client COM port
If you are willing to use a crappy web browser that lets you run ActiveX, you could develop or try to find an ActiveX that allows such connection.
ActiveX can do virtually whatever it wants on the client computer. There are some hoops to jump. Once you understand that this opens a vector of attack to the client, you will also understand why we have moved away from this technology.
Let me paraphrase that: DANGER! DANGER! DANGER!
Now, why do you need to do this from the web?
If you need to access the client COM ports, the only kind of legitimate reason I can think of is that you trying to avoid installing software. If that is the case - in my opinion - you are doing things wrong.
If not altering the software on the machine is a workplace policy and you need this for your work. I would have to advice to follow the bureaucracy of your company.
If for whatever technical reason you cannot alter the software on the machine, try booting with live CD or USB that allows you to load the tools you need.
Are you sure you cannot use a different machine to access the device you need to communicate to? Perhaps one without the software restriction. I know we could be talking about buying another machine, perhaps you could do with something small, like an Arduino or a Raspberry Pi.
Ok, stop trying to hack people already.