While there is no online edition of The Observatory, most content is available at ADS. In particular, can one retrieve all but the last couple of years in reverse chronological order or in chronological order. Note that all book reviews are contained within one item, but otherwise there is a one-to-one correspondence between what is in the Magazine and what is at ADS. (Note that some more recent items show up if they have been cited or posted to arXiv, though the source won't yet be available from ADS.)
If you know the volume and page number, then one can retrieve the scanned article directly. For example, you can access my review of a book by the late, great John D. Barrow if you know that it is in volume 133 on page 232. (You can select The Observatory from the list of journals, but of course other journals as well).
To find items by a particular author, use "bibstem:OBS" in addition to the "author" tag in the ADS search form, e.g. bibstem:OBS author:"Helbig, P" shows contributions from me, but excluding my book reviews and things from the past couple of years (see above).
To select just a given year, volume, or items starting with a given initial page number, use the somewhat confusingly named "paper form", (put "OBS" in the "Publication" field) e.g. one can retrieve all of volume 135. (Note that I changed "desc" to "asc" in the URL generated by the query so that the items appear in the same order as in the paper Magazine.)
Thus, specifying just "page:1" will return a list of volumes. That works because each volume starts numbering at page 1 and that is not reset for each issue. The PDF file and the "scanned article" will both be just the first item, however with the latter can navigate continuously forward via the "next article page" link. By going to the last page in the left-hand sidebar, one can navigate via the "next article page" link to the next item and, and from the last page of that to the following item, and so on until at the at the end of the last item, one can navigate to to the first item in the next issue in that volume (but not from the last page of one volume to the first page of the next). Note that there is one volume per year except that the years 1941–1946 have two years per volume.
Note that in the years 1886–1920, the actual Magazine had an Annual Companion, which starts out each volume, so one must first navigate to the end of that then use the "next article page" link to get to the beginning of the Magazine proper. (Sometimes, one must go through a few "next cover page" links before encountering the "next page" link which takes one to the start of the regular issue, but that has the wonderful side effect of presenting some turn-of-the-century advertisements.)
Of course, the methods above work modulo bugs at ADS, which one occasionally runs into.