This may be better suited for the Project Guidance section of the forums. But to answer your questions..
Yes, depending on your definition of HD. You wont get DSLR quality, but there are plenty of camera modules for the Arduino
The best option? That depends on what you want it for. Short answer: probably not. If I had a better idea what you were planning, I could give a better answer.
The camera modules are fairly plug-and-play. There will be a bit of coding and wiring, some positioning of the camera, etc. to get something that you might want to use. For all things Arduino, it's pretty open source and you can find code and wiring diagrams online for just about everything you could want.
There are several smaller Arduino boards such as the Trinket or the Pico, but you run into issues of needing more pins to do what you want.
I don't think a pico has enough pins to use that camera - you may be stuck with the Nano (I don't claim to know every board out though, there may be some with enough pins). If you want to save you photos you will need an SD card reader. You may also want a push button to tell the Arduino when to collect an image. That can also be done over a serial connection (USB to a computer) but then you're tied to a computer that probably has a webcam built in. If you're going the portable route, you will need a battery that supplies between 6 and 20V (but 7-12V is recommended). If you want to get fancy (and who doesn't), a 3D printed case can also be made.