I mounted a vinyl picture window sideways, can I drill small weeping holes into the frame?
You will certainly need to provide for some drainage of the now "lower" track of the window so that accumulated water can flow out. 

One thing to also consider is the overall design of the window itself. Many window units are designed with a definite UP and DOWN side to them. When you rotate such window by 90 degrees certain aspects of the window may be less than optimum for proper operation of the window. In particular with a window designed to be opened you may need to adapt certain aspects such as locks, latches and mechanisms designed to hold the window open. In the worst case a slide for the movable part of the window which was on the side now ends up on the bottom. Accumulated dirt in the track could make the window hard to open. The exterior top edge of some modular windows are also designed with a upper water drip edge which you may now have to substitute with a separate part in the installation.