It will depend on what level of web developer you are referring to. There is no way you can pass yourself off as a "professional" web developer in 3 months.
A professional is expected to know multiple back ends, frameworks, languages and web APIs which will take a lot of time to learn.
There is also the aesthetics (artistic details) to consider. In a large web production operation they may employ graphics designers to do the initial design work and use developers to impliment the design. A smaller operation may require web designers with graphics design skills also, and In a one man operation you have to do it all. That means knowledge of colour theory and graphics design skills.
So three months to a year is fine for basic web work but you're unlikely to be competent enough to write an online banking system in that time. It's a matter of what level you are aiming for.