If you go back to the Elements of Euclid (300 AD), mathematical formula were described through full sentences, that were unambiguous by construction, like "add 3 to 5 and then multiply by 4".

Around the XVIth and XVIIth century, we see the formalisation of symbolic calculus, and with that the ambiguity we're now used to: does "a + bc" means "(a+b)c" or "a + (bc)"? To ensure unambiguity, delimiters were added. I've used the parenthesis on my example, but this was only one of the multiple delimiters used in the early days. Notably:

• Using underline or overline, called vinculum. Delimiters through overline can still be seen in today's square root notations, where the overline determine where the square root stops.
• Using points instead of parenthesis, Descartes for example would write ".a+b. c" for "(a+b)c".
• Spacial notations, like putting "a" and "+b" in column then followed by a curly bracket "}" and a "c" to describe "(a+b)c". Also used by Descartes, for example in its Géométrie book. Some spacial notations survived up until today, like using exponents or fractions.

However, even in the early days, authors did not write explicitly all the delimiters. In a lot of cases, only one interpretation made sense because of consistency of units (if a is an area and b and c are distances, then "a+bc" obviously means "a+(bc)" because otherwise you would be summing a distance with an area). When there is an ambiguity, if one of the two possibility is absurd, then there is no ambiguity, so no reason for parentheses.

In cases where there was no unit, ambiguity was still a possibility, but scientists still wanted to simplify their expressions and use the minimal number of symbols. Then, practice made the rules. It is a fact that expressions of the form sum-then-product (so "(a+b)(c+d)") are rarer in practice than expressions of the form product-then-sum (so "(ab)+(cd)+..."). One reason being that you can always expand the first one, but not always factor the second. So scientists naturally removed the delimiters in the cases that were the most frequent, and kept it in exceptional cases.

Note that in the modern era, something did shook PEMDAS conventions: early calculators and computers. Indeed, for technical reasons, they couldn't use the usual PEMDAS, so a lot of alternative solutions were created. Notably:

• The naive order of operation, which simply goes from left to right so "3+4*5" is "(3+4)*5" but "4*5+3" is "(4*5)+3".
• The reverse polish notation, which is a postfix notation, so "3+(4*5)" would be typed as "3 4 5 * +" while "(3+4)*5" would be typed as "3 4 + 5 *".

Up to my knowledge, the technical restrictions disappeared too quickly to have a significant impact on the way we use order of operations, though the fact that some calculators used the naive order of operation probably confused a lot a students when they had to use PEMDAS in their exams.

Main source: La révolution symbolique - La constitution de l'écriture symbolique mathématique