I want to know if you can still get it without using any sort of hack.
Yes, but you will need Pokemon Stadium and a transfer pak. A strong team of 5 pokemon and a pikachu on your pokemon yellow cartridge is strongly recommended (because you can't use a rental/registered team on stadium).
Now do the following:
1) Get to round 2 (prime cup, master ball division) using any pokemon.
2) Now enter the tournament and select pikachu and your other 5 pokemon from yellow.
3) For every fight, select pikachu. You don't have to use him at all, just have pikachu on your team every round.*
4) After beating the round 2 prime cup** (master ball div) you'll get a small cutscene before confirming which move to replace with surf.
Note*: Since pikachu is practically useless in the fights, you need to have really good pokemon that can basically 2v3 the opponent.
**Note****: You don't have to be flawless in the tournament, you're allowed to lose a battle
It is still possible to obtain a Surfing Pikachu in the first generation.
You will need the following:
- A Nintendo 64
- A Transfer Pak
- Any first-generation game (Red, Blue or Yellow) with a Pikachu
- The first Pokémon Stadium game in the same language as your cartridge (note that for Japan, it is a different game)
Make sure all batteries are still working.
If your game is Japanese:
- Beat the Master Ball division of the L1-30 division with a Pikachu in your party. You will then be allowed to teach that Pikachu Surf.
If your game is not Japanese:
- Beat the Master Ball division of the Round 2 Prime Cup using only Pokémon from your cart without saving (continuing is OK), and have your Pikachu participate in the final battle. You will then be allowed to teach that Pikachu Surf.
You need to use Pokemon Stadium for the N64.
After you beat the R-2 Prime Cup Master Ball Division with Pikachu selected to go into every battle, you will see a scene of Pikachu in front of a sunset that will say "You Won Using Pikachu" and then it will ask to select a move to be replaced with Surf. Congrats.
This is backed up by Bulbapedia
A Surfing Pikachu minigame, Pikachu's Beach, is included. However, to play this the player will have to first teach Pikachu Surf, which requires the use of Pokémon Stadium.