Is there another way to trek into Machu Picchu without a permit?

No, the permitted Inca Trail is the only way to arrive on foot through the Gate of the Sun at Machu Picchu. You can trek on other Inca Trails that get you very close to the site.  Only those on the Inca Trail (which includes the One-Day, Mollepata, and Classic trails) will trek directly into Machu Picchu. Other treks that take you close to Machu Picchu include Apu Salkantay, the Lares Valley, and the Sacred Valley treks.  These treks include a short train ride to Aguas Calientes and a bus ride to the site itself.