Phase 1 of this study, utilizing a rolling 6 design, will be conducted to determine a maximum tolerated dose (MTD) and recommended Phase 2 dose (RP2D), and to describe the toxicities of lenvatinib administered in combination with everolimus once daily to pediatric participants with recurrent/refractory solid tumors. Phase 2, utilizing Simon's optimal 2-stage design, will be conducted to estimate the antitumor activity of lenvatinib in combination with everolimus in pediatric participants with selected recurrent/refractory solid tumors including Ewing sarcoma/peripheral primitive neuroectodermal tumor (pPNET), rhabdomyosarcoma, and high grade glioma (HGG) using objective response rate (ORR) at Week 16 as the outcome measure.