Yes - You can now find the Zip Codes in your Google Docs sheet using the following URL:
Step 1: In this landing page, enter the address which you would like to search.
Step 2: I searched for the "1245 5TH ST" address in Washington, DC and the URL looked like this:
https://tools.usps.com/go/ZipLookupResultsAction!input.action?resultMode=0&companyName=&address1=1245+5TH+ST&address2=&city=Washington&state=DC&urbanCode=&postalCode=&zip=
Step 3: Copy and paste the following formula and replace the CELL, CITY, STATE:
=Mid(importHTML(CONCATENATE("https://tools.usps.com/go/ZipLookupResultsAction!input.action?resultMode=0&companyName=&address1=",CELL,"&address2=&city=CITY&state=STATE&urbanCode=&postalCode=&zip="),"list",15),Find("DC",importHTML(CONCATENATE("https://tools.usps.com/go/ZipLookupResultsAction!input.action?resultMode=0&companyName=&address1=",CELL,"&address2=&city=CITY&state=STATE&urbanCode=&postalCode=&zip="),"list",15))+2,11)
Remember that the URL in Step 3 should match the one from Step 2. The Cell will be dynamic based on your target Cell Value.
The function in Step 3 extracts 11 characters after the STATE value. Hence, you will have to tweak the MID function so that you get the correct values. This worked perfectly for me.
One last thing: Google allows only 50 Import functions per sheet. So, with the above method you can find only 25 Zip Codes in one go.