Imal nailed it; the antidote to VX (and Sarin and many other nerve gases) is the liquid form of a nerve agent called atropine.
Ideally (yeah, like there is an "ideal" to an exposure to a weapon of mass destruction) 2mg of liquid atropine is injected into your thigh every 6 minutes after exposure, or if the injection is "late to the party", injected directly into your heart.
For either, what you will probably want to possess is a NAAK (Nerve Agent Antidote Kit), but civilians will probably have a difficult time getting them.
But don't fret. Atropine in diluted solution and even topical cream form can be purchased legally for your pet (you do have a pet, right??). Without question having an atropine cream covering all of your skin would yield some level of protection (post-exposure) to VX or Sarin non-inhaled exposure (though lets not go trying this for fun, shall we).
To purchase, Click Here - or find anther such pet shop
Do beware, however, as atropine is itself extremely toxic and can kill you. It's one of the active ingredients that makes belladonna poisonous.
Atropine has medical applications other than as a treatment for nerve gas exposure, but the doses are usually in the micrograms, and messing around with it can be very hazardous to your health, causing temporary symptoms such as mydriasis, tachycardia, decreased salivation and sweating, diminished bowel sounds, urinary retention, hypertension, vasodilation, anxiety, disorientation, hallucinations, hyperactivity, and convulsions or coma. In practice, however, an adult has to really overdo it to risk permanent death.
The NAAK instructions rather dryly state: "If you are told that your pupils are getting very small or if you are having trouble breathing and your chest feels tight, use the atropine nerve agent antidote kit (NAAK), Mark I". Long ago, they used to tell you that nerve gas smelled like almonds, but that's not always a reliable indication of exposure, and I guess they stopped saying this after a few paranoids stuck themselves unnecessarily.
The subsequent warning states: "If within 5 minutes after the administration of any set of injections your heart beats very rapidly AND your mouth becomes very dry, DO NOT give yourself another set of injections". Those are some of the symptoms of atropine overdose.
There are other antidotes that are used to improve resistance to a nerve agent attack. In a NAAK/Mark I kit, both an atropine autoinjector and a pralidoxime chloride (2-PAM Cl) injector are included. The 2-PAM is an important counteragent for fast-aging nerve agents such as Soman (GD). The Army also uses pyridostigmine supplementation in tablet form for such agents, issued as the Nerve Agent Pyridostigmine Pretreatment (NAPP) Tablet Set. And there are others.
I believe it is currently possible for civilians to obtain a NAAK by prescription, but good luck getting a doctor to go along with that. If a nerve gas attack occurs, however, I expect that to change instantaneously.
The government is, of course interested in protecting us from ourselves as much as from terrorism, and, frankly, for every x number of NAAKs in civilian hands, y number of people will use them improperly and wind up in an emergency room. Some may die.
But I am hopeful that, ultimately, the paternalistic stance of restriction imposed on us by the feds will be relaxed. This should happen by necessity, if nothing else.
Stay safe.