you can get any type does not change anything wot your starter Pokemon is
^ That is false I take it you are playing leafgreen or firered, so the correct answer is the roaming dog that has the type advantage over your start that being suicune
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The legendary dog which you face will alway be typed to be at an advantage against your starter. Therefore, if you started with Squirtle (which becomes Blastoise), then the dog you will face is Raikou. (Bulbasaur=Entei; Charmander=Suicune)
To get a legendary dog, you must first obtain the National Dex. Once you ave it, the dog that appears is determined by your starter. If you chose Charmander (fire), a Suicune will appear. If you chose Squirtle (water) a Raicou (sorry about spelling) will appear. If you chose Bulbasaur (grass) an Entei will appear. The dog will be roaming around the game in random fields. Once you see it, it's current area will be shown in the pokédex. Do not use fly to try and get it, as the legendary dog moves as far as possible from you when you use fly.
I've done research on this question myself and the answer is YES! After you beat the Elite Four you will be able to find the legendary dogs randomly in the grass (just like in Gold and Silver). However, for your game you will only be able to catch one of the dogs, which dog it is depends on your starter pokemon. The determiners are as follows: Starters-Legendary Dog: Bulbasaur-Entei Charmander-Suicune Squirtle-Raikou Good luck to you!
Charmander
If you chose Bulbasaur, you'll find Entei. If you chose Squirtle, you'll find Raikou. If you chose Charmander, you'll find Suicune.