I am trying to find an MCU in which there are some specific requirements for I/O peripherals (such as UARTs, SPI, I2C, etc). I also want to be able to search by the footprint size. Should I go to each popular MCU website and search from their selection list or is there any other possible way to do that?
The strategy would indeed be to go to the various manufacturer web sites and use their parametric search engines to narrow down the selection to 4 UARTS. Each manufacturer will have a slightly different search facility so you have adapt to that.
One benefit here is that the number of main MCU manufacturers has been reduced in the recent decade due to companies either joining forces or some exiting the market.
For small footprints it can often be useful to consider the SiLabs1 offerings. The Leopard Gecko series of ARM core MCUs offer five UARTs in QFN64 or QFP64 packages.
1No affiliation with company but user of their products.
You can also use individual manufacturer's sites, which will likely have more parts available, but distributors like those linked above sell parts from many different manufacturers, and allow you to directly compare parts from different manufacturers as well.
I don't think there are such website, at least they will not be supported by major brands, since it will show both their advantages, but also disadvantages compared to other MCUs.
One MCU that has 4 UARTS is the ATmega2560, known from the Arduino Mega 2560, which has 4 UARTs. (see also the comment of Marcus Miller below).
ST.com also has many MCUs having (at least) 4 UARTs, and it has an easy search tool which can be found here: ST MCU Finder. You have to install it though (there is also an Android app which can do the same, with the same name).