I am looking for an article that was published in either Rock and Ice or Climbing Magazine around 1999. I know the area and route of the article topic as well as the author's name. I may even have a copy of the cover image and/or and image used in the article.
Are there digital back issues of the magazine? Is there a search function? Are there libraries that would have back issues in print (or microfilm)?
The site Archive.Org has copies of magazines that had internet versions. A search for "Rock and Ice" produced a magazine called Rock and Ice Climbing.
There is a possibility that your article could be there.
There are indeed libraries which have print or microfilm copies of those publications. San Francisco Public Library has a Periodicals floor where they can be accessed. They appear to have issues since 1995 (Climbing magazine) and 1998 (Rock and Ice magazine).
There are services which provide online versions of back-issues of these magazines, too. One such service is RB Digital Magazines, which again one can access via their online San Francisco Public Library membership.
That particular library provides services and membership (library cards) to anyone who shows up in person at a branch with proof of residence in the state of California. One doesn't have to be a San Francisco resident.
Since the San Francisco library provides these materials and online access, the answer to your question is "Yes", what you're asking for exists, and if this particular library doesn't suit you, I'm sure the same materials and services could be found at other libraries too. San Francisco is particularly modern and well-funded, so, libraries with similar budgets or endowments would probably be the best places to try. So, I'd suggest investigating public libraries in major cities, or, campus libraries at larger universities.
