The only way I've found to stop something from using Sticky Web is either Prankster Taunt or Magic Bounce, but Magic Bounce is not reliable since the only Pokemon that have it are all weak to Electric/ Bug. Magic Coat would work too, but most of the Pokemon that can learn it are part psychic type. If you really wanted to, then you could use a Magnezone/ Magneton with Magic Coat, but to be honest they have better things to do, and after the first turn, it's really hard to predict when your opponent will switch to attacking instead of using Sticky Web.
Sticky Web will only affect things that touch the ground, so you could just let it happen and use anything with Levitate/ flying type/ holding an Air Balloon. Anything with Clear Body is also unaffected, and anything with Contrary will have their speed raised instead.
Sticky Web is also removed by Rapid Spin & Defog.