Answer 1

Nope, you cannot catch/obtain any of the Eeveelutions (or Eevee itself) in the original B/W by regular capture from the wild. However, Eevee and it's evolutions can be seen by battling the following trainers:

• Lady Elizandra, who can be found in the Royal Unova. (Eevee)
• Ace Trainer Lucielle, who can be found in the Nimbasa City Sports Dome. (Vapoureon)
• Veteran Arlen, who can be found in the Nimbasa City Sports Dome. (Jolteon)
• Backpacker Patty, who can be found in the Nimbasa City Sports Dome. (Flareon)
• Ace Trainer Junko, who can be found at Route 14. (Espeon)
• PKMN Ranger Heidi, who can be found in the Nimbasa City Sports Dome. (Umbreon)
• Waitress Bonita, who can be found in the Nimbasa City Sports Dome. (Leafeon)
• Veteran Sayuri, who can be found in the Nimbasa City Sports Dome. (Glaceon)
  You may also, of course, trade, migrate or transfer them to your game.

Source: Various Serebii 'Trainerdex' pages:
http://www.serebii.net/pokedex-bw/trainers/133.shtml
http://www.serebii.net/pokedex-bw/trainers/134.shtml
http://www.serebii.net/pokedex-bw/trainers/135.shtml
http://www.serebii.net/pokedex-bw/trainers/136.shtml
http://www.serebii.net/pokedex-bw/trainers/196.shtml
http://www.serebii.net/pokedex-bw/trainers/197.shtml
http://www.serebii.net/pokedex-bw/trainers/470.shtml
http://www.serebii.net/pokedex-bw/trainers/471.shtml

Comments

If you have black or white 2 you can just trade an eevee from there, I don't think you have to see an eevee before doing this, but I would just in case.

Answer 2

You cannot get eevee or its evolutions in black.You have to trade from someone or poketransfer from previous generations.

If you want to see eevee or its evolutions,these sources should help you indicating which trainer has those which should help you fill up the pokedex.

Eevee
Vaporeon
Jolteon
Flareon
Espeon
Umbreon
Leafeon
Glaceon

Hope this helps!