Two questions to ask yourself:
- What would happen if the web design client found out who your employer was?
- What would happen if your employer found out about your web design client?
Most likely, if you think this through, you will find that you would be placed in a bad position in either case. If either party finds out about this connection, they are likely to think that you took advantage of your position at your job to make this connection (even if that was not the case). This will reflect poorly on you.
And if you don't want both sides to find out about each other, that would place you in an untenable position of trying to maintain secrecy.
I would avoid this.
- Even if you aren't doing anything wrong at all, it will look wrong if someone finds out (and it will be quite hard to prove that you didn't do anything wrong).
- Once you got into this situation, you might be placed into a position of greater conflict of interest (for example what if you start doing web design for company X, then your employer asks you to be directly involved with company X for them?).
- If you do have any "insider" knowledge of the company, from either side, it will be hard not to take advantage of that in your other role.
Also, your question claims you will not take advantage of the company's relationship with the supplier at all. If so, there is no real benefit to approaching these particular companies--just cold call someone else.