Embed Form In Header?

You can place a form wherever you want if you use the function call in your theme template. So there is no reason why you couldn't do this.

Just keep in mind when using the function call to display a form you need to also enqueue the necessary scripts using the Gravity Forms enqueue function because it can't do it automatically because it may be executed from a location where it can't do so.

Documentation for embedding a form, including via a function call can be found here:

http://www.gravityhelp.com/documentation/page/Embedding_A_Form

Another option is to modify your theme to add a sidebar in the header and then use the Form widget to display a form in that sidebar.