Why don't they take a picture of every animal that comes into the shelter and posted on the website so we can see if our animal has been picked up or if there's one we would like to adopt?

Best way for this to happen is to have volunteers. You have to consider that you are able to get dogs and cats for very cheap and sometimes even $5 or $10. The turnover is very quick but yet it is not quick enough to save all the animals so there is likely just not enough manpower or time.

2 things, 1) i believe they have another organization actually managing their site so have little control. 2) I'm guessing there is almost always more work to do than people to do it, but if someone is willing to volunteer to come every day to take the pictures they would appreciate it very much. PS. I do not work there so my info is hear say.

Because of the vast number of animals that go through that place. They already need all the staff they have. I doubt they could spare the manpower to set that up and maintain it.