Abstract: Laser wakefield acceleration (LWFA) may achieve TeV/m gradients using high-density solid-state plasmas as accelerating media. However, the application of bulk solid materials requires attosecond laser pulses, such as X-ray lasers, to drive wakefields at these high densities. Additionally, the short wakefield wavelengths associated with solid-state plasmas greatly limit the accelerating length. An alternative approach employs 2D carbon-based nanomaterials, like graphene or carbon nanotubes (CNTs), configured into structured targets. These nanostructures are designed with voids or low-density regions to effectively reduce the overall plasma density. This reduction enables the use of longer-wavelength lasers and also extends the plasma wavelength and the acceleration length. In this study, we present, to our knowledge, the first numerical demonstration of electron acceleration via self-injection into a wakefield bubble driven by an infrared laser pulse in structured CNT targets, similar to the behavior observed in gaseous plasmas for LWFA in the nonlinear (or bubble) regime. Using the PIConGPU code, bundles of CNTs are modeled in a 3D geometry as 25 nm-thick carbon tubes with an initial density of 10^22 cm^-3. The carbon plasma is ionized by a three-cycle, 800 nm wavelength laser pulse with a peak intensity of 10^21 W cm^-2, achieving an effective plasma density of 10^20 cm^-3. The same laser also drives the wakefield bubble, responsible for electron self-injection and acceleration. Simulation results indicate that fs-long electron bunches with hundreds of pC charge can be self-injected and accelerated at gradients exceeding 1 TeV/m. Both charge and accelerating gradient figures are unprecedented when compared with LWFA in gaseous plasma.