So can one read Punch today? Of course, this is the digital age!
The Online Books Page has a serial archive of Punch from 1841 to 1922. Once you select an issue there is an icon slideshow of the pages of the issue and options to read online, or download, including an download specially formatted for the kindle. While the years are incomplete there is a tremendous amount of material here not only the aforementioned cartoons, but hilarious descriptions of such Victorian icons as the Crystal Palace and essays lampooning Victorian life and politics.
In addition, while it is no longer in print, Punch has it's own website with a searchable database of cartoon images from 1841 through 2002. While it's unclear to me how complete this database is, if you're only interested in the cartoons it's a little easier to wade through than the digital archive.