Is it possible to achieve this without log in to the application and
saving values to the database and without breaching the security.
When the manager gets an email & he clicks on the approve. Ideally a request is fired from manager's system carrying an authentication token or something like a remember me token. In this case the application won't prompt the manager to login. It will tally the request token with token stored in the db & would let manager in.
Check how remember me works in web applications. You can create a remember me token for manager for an indefinite amount of time. This way the app will never prompt the manager for a login. But there is a caveat, this approach is a little vulnerable for attack. If there is a man in middle attack the hacker will get hold of the manager's client side cookies & would always login with ease. The auth token should be changed periodically. You can always implement remember me, it just depends on what level of security you require.
Bottom line: Yes it is possible, generate a remember me token which will always be included in the manager's email request for approving employee's leave request. Just follow the best practices to implement a remember me token.
Another way is
If your application is implementing security using filters. You can bypass the request for leave approval. Just put a secure none attribute for that particular request in ant style say "yourapp/approval" resource. Now the server side code would let all the requests in this pattern pass without security check. But this approach is a strict No No. If this happens a hacker which intercepts the leave request from the employee's browser can approve his leave by himself. No need of manager's authentication.
If yes, how can i implement this?
Implementing remember-me for a website
You can also integrate Spring Security with your code. It's hell of a framework, one stop solution for all the security related features which your app would require. You don't have to write the login for implementing auth token & stuff from scratch.