Of course each payor is different but here is my take on it. There is no "shared" component to the work of giving two shots so the 2nd shot fee should not be considered bundled into the other and therefore 59 would not be needed. You only need it if it would be common to bundle the two codes together.
The modifier 25 on the EM may be needed depending on the payer. Some payors bundle the shot admin code into the EM. So if you are reporting an EM code in addition make sure it is indeed "separate" and "significant" and if so by all means append the 25.
HTH