Is there a way to embed a Google Docs form in a page without using plugins?

No, that’s not possible. You can find a list of providers for embeddable coontent in wp-includes/class-oembed.php:

'#https?://(www\.)?youtube.com/watch.*#i'            
'http://youtu.be/*'                                  
'http://blip.tv/*'                                   
'#https?://(www\.)?vimeo\.com/.*#i'                  
'#https?://(www\.)?dailymotion\.com/.*#i'            
'#https?://(www\.)?flickr\.com/.*#i'                 
'#https?://(.+\.)?smugmug\.com/.*#i'                 
'#https?://(www\.)?hulu\.com/watch/.*#i'             
'#https?://(www\.)?viddler\.com/.*#i'                
'http://qik.com/*'                                   
'http://revision3.com/*'                             
'http://i*.photobucket.com/albums/*'                 
'http://gi*.photobucket.com/groups/*'                
'#https?://(www\.)?scribd\.com/.*#i'                 
'http://wordpress.tv/*'                              
'#https?://(.+\.)?polldaddy\.com/.*#i'               
'#https?://(www\.)?funnyordie\.com/videos/.*#i'      
'#https?://(www\.)?twitter.com/.+?/status(es)?/.*#i' 
'#https?://(www\.)?soundcloud\.com/.*#i'             
'#https?://(www\.)?slideshare.net/*#'                
'#http://instagr(\.am|am\.com)/p/.*#i'               

Google is not in that list, so you have to ask the administrator.