Contests and saving throws do not require an action by themselves, so a surprise round does not take that ability away from a target.
Unfortunately, there really is no mechanical advantage given to you when you sneak-grapple, other than getting one extra turn to attempt the grapple. Given the circumstances, your DM could allow some sort of advantage (the mechanical advantage or the literal advantage) to these types of checks, but it would not be RAW (and that's okay!).
More importantly than that, you're attempting to role-play a situation and this should be encouraged. See: Are melee combatants limited to standing around saying “I attack”? Talk to your DM about it to see if they're okay with this type of behavior.
If you want to create a hostage situation without a grapple check, you can choose to knock a creature unconscious when you reduce it to zero hit points with a melee attack, even when using your sneak attack. This will ensure they are unconscious, but not dying. You can then carry them, and threaten their unconscious body.
Knocking a Creature Out. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable. (PHB p.198) [emphasis mine]