Starting Items:
Corrupting Potion is excellent for trading and sustain.
Lane Items:
Doran's Rings are great if trading with targets that are trying to push and/or engage on you. Can get anywhere from none to three, although I'd rarely get more than two, depending on how much mana sustain you need.
Bami's Cinder is great in conjunction with Doran's Rings for trading and pressuring against most melee opponents.
Ninja Tabi are the best boots to have against any lane opponent that hits you with auto attacks.
Catalyst of Aeons is an excellent alternative to Bami's Cinder and Doran's Rings for mana and health sustain. Build it against ranged pressure, particularly lanes you're having trouble engaging on for trades.
Everfrost can be built out of Catalyst of Aeons vs kiting opponents. It'll help you survive and connect with some serious burst damage.
Item Options:
Sunfire Aegis is a great split-push item on Wingborne Storm Udyr. Build it if you built Bami's Cinder.
Trinity Force is another excellent split-push item for Udyr. It provides him with incredible 1v1 damage, mobility, and stat distribution.
Iceborn Gauntlet is an interesting alternative to Trinity Force. It's much less potent as a split push item, but if you decide to group with your team instead, can provide quite a lot of utility in fights, and is excellent for helping catch people.
Spirit Visage is very strong on Udyr. Iron Mantle makes great use of the passive, and the CDR is appreciated.
Dead Man's Plate is an excellent armor item on Udyr. It helps him catch for engages, and move around the map more quickly.
Zz'Rot Portal is ideal later for split-push Udyr. It allows him to siege more quickly, and proxy-pressure while rotating between lanes.
Wit's End is the only other good damage item on Wingborne Storm Udyr, particularly with the sticking power of Iceborn Gauntlet, although it can work if you need more help fighting it out 1v1 while splitting.
Randuin's Omen is a solid buy if you're fighting against a crit based team, particularly if you're grouping.
Thornmail is excellent budget armor, and can help you fight it out vs certain auto attackers.
Warmog's Armor is a decent CDR option, and can sometimes be substituted for Spirit Visage if you're playing more of a rotation/kite based play-style or aren't concerned with magic damage.
Guardian Angel is a great late game item if you're grouping for fights.
Boots of Swiftness are your greed boots. Build them if you don't need Ninja Tabi and are splitting, vs heavy slow comps, or if you're snowballing and trying to chase people down.
Mercury's Treads are your best option vs medium to heavy CC comps. Udyr hates CC. These help. But Ninja Tabi are really strong right now.