Can you become a Jedi knight if you wanted to? Will the academy accept applications or they only choose for themselves?
It depends on the era.
Before the execution of Order 66, The Jedi routinely checked the children of the republic for acceptable levels of midiclorians (SWI:TPM). Once a candidate was identified the Jedi would offer to train the child and, if necessary, use the Jedi Suggestion to get permission. These children would be brought to the academy and trained from an early age so that they would not remember their parents or form an attachment.
After RotJ, Luke reestablished the order. In doing so he recruited many adults and even married couples into the new Order. It was no longer necessary for children to be pried from their parents' arms and the Order even provided lodging for families of young jedi. The new Order was willing to train anyone who showed any force talent at all, including many that likely would not have met the midichlorian count that was required by the original Jedi Order. As for where, most applicants came to the Jedi Temple on Coruscant. However most were still discovered by Jedi in the field though through the force rather than using midichlorian count.