Do I have to put ,Inc. every time I mention our company in a single article?

I write blogs for our company website, often commenting on what services our company provides, so I will usually mention our company name a few times in the article. It is a local company, but it is incorporated and the official name of the company includes ",Inc." at the end.

I always include the full name of the company in the first instance in an article, but because our full name is a bit long, I've been shortening it in the rest of the article. Is that bad grammar or improper etiquette?

Example: if our full name is "A-1 Safety & Training, Inc." can I say the full name once at the beginning of the article and then just say "A-1 Safety" in the rest of the article or should I use the full name in every instance? What is proper?

As everyone noted in the comments, this is a question of style, not grammar. I will here reproduce the recommendations of two influential style manuals: The Chicago Manual of Style (CMoS, 17th ed.) and The AP Stylebook (2019).

CMoS

10.24: Abbreviations and ampersands in company names

Abbreviations and ampersands are appropriate in notes, bibliographies, tabular matter, and the like. See also 14.135.

Ginn & Co.
JPMorgan Chase & Co.
Moss Bros.
RAND Corp.

In running text, company names are best given in their full forms. It should be noted, however, that some full forms include ampersands and abbreviations. If in doubt, especially with reference to contemporary firms, look up the company name at a corporate website or other authoritative source. Such elements as Inc., & Co., and LLC may be omitted unless relevant to the context.

Johnson & Johnson was founded in 1886.
JPMorgan Chase operates in more than sixty countries.
AT&T Corporation was once known as the American Telephone and Telegraph Company.

AP Stylebook

company names

… You must include the full company name somewhere in the story. This ensures the story will be among the search results on major websites.

The formal name need not be used on first reference — for example, Costco is acceptable for Costco Wholesale Corp. — but it should be contained in the body of any story in which the subject matter could affect a company’s business.