Only if you didn't buy the ammo upgrades for the suit. Also at max ammo upgrade you had 7 slots for the chainsaw so you would have enough for 7 full ammo refills from tiny enemies, so not having a recharge shouldn't be a issue unless you waste it. Plus like I said there was a ammo boost rune you could get, when mastered would even make BFG ammo RNG drops, which just strait up broke the game.
Of course I'm just basing this on my experience replaying the game last week in anticipation of this, but I never really payed attention to ammo yet I never didn't have ammo when I wanted to use a weapon, and always had chainsaw uses to spare, but I also got all ammo upgrades first before health and armor.
Also that's my only real gripe with ammo in this game, ammo upgrades for shells and rockets compared to bullets (unchanged from 2016) and plasma was a bit overkill.