Apple - How can I take a larger screenshot on my iPhone?
Natively available in iOS 13 and later
This functionality is now built into iOS 13 by Apple. Safari running in iOS 13 lets you screenshot the entire webpage as a single continuous image.
To do that, open the desired webpage in Safari, and simply take a screenshot as you would normally do on your device, and tap on the thumbnail shown immediately shown towards the bottom left.
You'll be presented with two options at the top in a segmented control:
Screen
and
Full Page
With the Full Page item selected, tap on the Share icon in the top right to share the capture via desired medium. AirDropping it for instance, exports a single PDF file, which can be easily exported into a desired image format.
Webshot Free can do this (there's a 99¢ ad-free version too). It's supposed to let you just change http:// to webshot:// and it will open the same URL in in the app. That didn't work, but cut+paste did.
There may be other apps which are better for certain needs, depending on what you want to actually do with the page. If you just want a JPG or PNG of it, Webshot Free works fine.
If you are trying to save a page for reading later, iCab Mobile ($2, universal) will let you save a .webarchive or a .pdf file. [Note: the print to pdf function is actually provided by 3rd party webapps, but it's integrated into iCab].
There are a variety of "Print to PDF" apps in the app store, PrintBureau, seems to be one of the better ones (and is universal).
This question has been asked on StackOverflow.
Webpage Capture ($2.39, demo, confirmed on SO) or Website Screenshot ($0.99) should do the trick.
I have not tested any of them.
As a side note: since Safari for Mac itself doesn't do full page export*, it really doesn't come as a surprise there's no way to do so on iOS.
*yes, of course you can print as PDF, but that's completely different from saving a full-dimensioned, not-paginated, not-styled-for-print version of the website (for which a third-party app such as LittleSnapper is necessary)