Record Static Charge Affecting Vertical Tracking Force (VTF)?

As dover alludes, yes it can. How much depends on several factors that can not be predicted like the magnitude of the charge.

1st off make sure the phonograph chassis is connected to HOUSE ground which may or may not be phonostage ground. You have to check with a meter. 

Use anti static record sleeves like the ones supplied by MoFi or Sleeve City. Paper sleeves are handily the worst.

Make sure your records are completely discharged before, during or after play. I use a conductive sweep arm that discharges the record during play like this one

There are much more expensive ones out there but they do not do the job any better and are so long they will not fit under a dust cover.

You want as low an impedance to ground as you can get. The carbon bristled record brushes would be great except the person holding the brush is not connected to ground and even if he was his impedance is several megaohms. To make these work drill and tap a small hole or you could drill and use a self tapping  screw. and use a long wire to connect it to house ground then swipe the record before and after play. This is way more effective than the ionizing static killers which I personally do not like. The sweep arm not only kills static but it also sweeps any incidental dust away from the stylus. As long as they are clean they will track right along with the tonearm. I clean mine with brake cleaning fluid once in a blue moon. 

I’m in the desert southwest and the current humidity in my house right now is 9%.  My floors are carpet.  One turntable has an acrylic platter, no mat, the other an aluminum platter with rubber mat.  I use a dry Discwasher brush before each play.  I don’t experience any static electricity when brushing, removing a record, or lifting the stylus.  I do build a slight charge on the trip to the equipment rack which I discharge by touching the metal post on the rack before I touch the cueing lever.

It seems counterintuitive to place the blame entirely on humidity.  Perhaps you have another issue which is overlooked?

Shure Corporation published a white paper on Static Charge as it pertains to vinyl, and they actually measured increases in VTF and correlated them with the amount of charge on an LP. I wish I could recall the exact results or find the paper on line right this minute, but I can’t. Suffice to say that VTF can be dramatically influenced by static charge, enough so that in some cases it would theoretically endanger the cartridge suspension. I believe that the major enemy that causes charge build-up is us. You can clean and store your LPs with meticulous attention to avoiding static charge. Then in one fraction of a second you can transfer charge to an uncharged LP just in the handling process that must precede playing an LP. Wool carpets, type of footwear, etc, could play a role. Like someone else said, ground yourself before touching the LP. Also, I agree, avoid paper sleeves; the act of sliding an LP out of a paper sleeve can charge it up nicely.

Use Furutech Destat 2 as everything else is a joke.