Instructions
Go to Azalea Town and enter Union Cave on any Friday after you have defeated Morty. You need the Surf ability that his badge grants you in order to traverse through the cave. The Lapras only appears on Fridays, so do not bother going into the cave on any other day or it will not be there.
Surf to the south until you find a staircase leading down to the next level. Go down the stairs and battle the Pokemaniac you find there. Surf into the next area and fight another Pokemaniac. After you defeat him, head down the stairs.
Surf to the southeast to find a platform and a trainer. Defeat her and return to the water. Continue along the linear path to the right to encounter another trainer. Fight him and return to the first platform. Head south to find the last trainer in the area. Defeat her and go into the water.
Surf to the right of the final trainer to find a large area of water and a Lapras swimming around. Go up to it to engage it in battle. The Lapras is level 20, so it may be somewhat challenging if you fight it immediately after defeating Morty. Weaken Lapras as you would any Pokemon before catching it. Net Balls, Dusk Balls, Heavy Balls and Ultra Balls are your best bet for a successful capture.