Yasharn only disallows you from paying life in order to activate an ability. Sylvan Library is a triggered ability, so by the time you are asked to pay life, the whole ability is already resolving. So Yasharn does not interact with Sylvan Library.
Sylvan Library is a triggered ability, no spell is being cast or ability being activated. An activated ability is always listed as "Cost:Ability", the colon is very important syntax. Yasharn would interfere if it was listed like "Pay 4 Life: Draw a Card" then it would be different, but nothing here is being cast/activated, so there is no overlap.
Additionally, from Yasharn's Gatherer page additional rulings:
9/25/2020 Yasharn’s first ability can find just one basic Forest or one basic Plains card if you don’t have both in your library or if you don’t wish to find both.
9/25/2020 If a spell or activated ability has a cost that requires a player to pay life (such as the activated ability granted by Lithoform Blight) or sacrifice a permanent (such as Kazuul’s Fury or Spare Supplies), that spell or ability can’t be cast or activated. 9/25/2020 Other things may still cause players to pay life or sacrifice creatures, such as a resolving spell or ability.
Last one would seem to apply here, so yes, they can still pay life for the cards.
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Additionally, from Yasharn's Gatherer page additional rulings:
Last one would seem to apply here, so yes, they can still pay life for the cards.