Should I use "I'ld" and the like in official documents like motivation letters or is "I would" better?
Not "I'ld" as the proper contraction of "I would" is "I'd" (without an "L")
Except in very formal writing, contractions are acceptable. Your reader isn't going to reject a good letter just because it uses "I'd". However "I would" is correct and safe. If unsure you can use "I would".
In letters to employers, government officials, utilities, lawyers, accountants, university admissions staff, etc, and in academic essays, I would advocate almost complete avoidance of contractions.