There is no rule explicitly allowing or prohibiting you from using bearings on your robot. However, there are several rules governing what kinds of materials you can use and where you can source them from. The most important rules for VRC robots are R5-R7:
<R5> Robots may be built ONLY using official VEX EDR components, unless otherwise specifically
noted within these rules.
[omitted for brevity]
<R6> Official VEX products are ONLY available from VEX Robotics & official VEX Resellers. To
determine whether a product is “official” or not, consult www.vexrobotics.com. A complete list of
authorized VEX Resellers can be found at www.vexrobotics.com/find-a-reseller.
<R7> Robots are allowed the following additional “non-VEX” components:
[various exceptions listed here]
Since you mentioned that you were competing in VEXU, you have some additional freedom in parts sourcing, which is described in VUR2 and VUR3:
<VUR2> Teams may use any official VEX Robotics product, other than the exceptions noted below, to construct their Robot. This includes those from the VEXpro, VEX EDR, and VEX IQ product lines. To figure out if a product is “official” or not, refer to the www.vexrobotics.com website. The following products are not permitted under this rule:
[omitted for brevity]
<VUR3> Teams are allowed to fabricate their own unique components from the following additional
raw materials for each of their robots:
a. An unlimited amount of non-shattering plastic from the following list: polycarbonate, acetal
monopolymer (Delrin), acetal copolymer (Acetron GP), POM (acetal), ABS, PEEK, PET, HDPE,
LDPE, Nylon (all grades), Polypropylene, FEP.
b. An unlimited amount of composite materials, such as G10 (Garolite), FR-4, or carbon fiber.
c. An unlimited number of plastic 3D printed parts.
d. An unlimited amount of steel and aluminum.
The intent of <VUR3> to encourage teams to explore fabrication techniques like milling, 3D
printing, injection molding, sheet metal punching, etc., to develop their own new robotic
components in addition to the “standard” set of VEX components permitted by <VUR2>. To
utilize these techniques, raw materials from the list provided in <VUR3> may be used.
However, the intent of <VUR3> is not to legalize all commercially available items made from
these materials. The only commercial components (other than pneumatic components) that
may be used are those purchased from VEX Robotics, as specified in <VUR2>.
For example, aluminum billet may be used to machine a custom bracket. However,
purchasing a custom aluminum bracket is not within the spirit of this rule.
Similarly, pre-drilled or extruded metal, such as angle aluminum, is not permitted, unless it
can be found on www.vexrobotics.com.
So, any old bearing you could buy at a hardware store would not be legal for use on your robot. However, a bearing that is an “official VEX Robotics product” available at vexrobotics.com would be legal. I’m not sure if such a part exists, but if it did I suspect it woudl be in the VEXpro line (thus, legal in VEXU but not VRC). Alternately you could design and build your own bearing using the materials allowed in <VUR3> and/or <R7>.