You actually do this is one step: send a POST requests to the information you need. For the first example, you would send these POST data:
__VIEWSTATE=dDwtNzQ5MzEyMDI3Ozs%2BHWC8LXZfQQTCTJWtmudNLStjn%2Fk%3D&txtLastName1=test&txtFirstName1=&txtBusinessName1=&txtLastName2=&txtFirstName2=&txtBusinessName2=&txtLastName3=&txtFirstName3=&txtBusinessName3=&txtLastName4=&txtFirstName4=&txtBusinessName4=&txtLastName5=&txtFirstName5=&txtBusinessName5=&cmdSubmit=Search
So you can use PhantomJS, as suggested, with a POST request and take a screenshot. Or you can use a web service to create the screenshots. Unfortunately, very few allow POST requests with custom POST data. Browshot (see API) lets you send POST data and use a custom Referer. So your screenshot requests would include (after you encode the POST data):
url=http://exclusions.oig.hhs.gov/search.aspx&referer=http://exclusions.oig.hhs.gov/search.aspx&post_data=__VIEWSTATE%3DdDwtNzQ5MzEyMDI3Ozs%252BHWC8LXZfQQTCTJWtmudNLStjn%252Fk%253D%26txtLastName1%3Dtest%26txtFirstName1%3D%26txtBusinessName1%3D%26txtLastName2%3D%26txtFirstName2%3D%26txtBusinessName2%3D%26txtLastName3%3D%26txtFirstName3%3D%26txtBusinessName3%3D%26txtLastName4%3D%26txtFirstName4%3D%26txtBusinessName4%3D%26txtLastName5%3D%26txtFirstName5%3D%26txtBusinessName5%3D%26cmdSubmit%3DSearch
I have tested it, and do get a screenshot of the results, like if I filled out the form and submitted it.