There are only a few cheap options for vegans in Japan with everything having some sort of non-veg thing inside it.
But, if you are ok to give up vegan-ism for a day or two and turn to eating milk products or may be go up to eating eggs as well then I can tell you many options.
As is the case with many people who switched from eating non-veg foods like meat/fish and all, to being a vegan they are still ok to some extent to eating milk and egg products, I will continue.
So, there is this Mexican restaurant called Guzman-Gomez in the ever happy street of Harajuku, here you can order your very own vegetarian Quesadillas, Fajitas or even Tacos by opting for the vegetarian option. They have cheese and some sauces inside which sure have eggs and milk inside them, but sure its a great option for vegetarians or eggitarians. The dishes are only about a 1000 Yens (10$) and you can get your stomach full by having just 1 Quesadillas.
As you must already be knowing that there are many many Indian restaurants and most have atleast 1-2 curries which are vegetarian friendly. But Naan is made up of egg and milk and curries also use some cream so for my Vegan friends, you can just request the waiter to not add cream to your curries and replace Naan with rice, that would be a great spicy Vegan dish for you!
Another one that I discovered recently is MOS burger. They are not Vegan friendly (or may be they are since I did not ask them about all the ingredients), but one thing is for sure that they are vegetarian friendly.
There are some burgers that allow you to replace the patty with Soy patty and it makes it a vegetarian dish. It is mentioned in small brown letters under the picture of the dish in the menu
They have some vegan options in the form of Kinpira burger which is a rice burger. Also, I will update here when I get to know that the Soy patty burgers are vegan or not but for now I am sure they are atleast a good vegetarian option.
Then there is Coco Ichibanya (Curry House) where you can find a Vegan menu (provided you are in a foreigners friendly area of Shinjuku, Shibuya, Aoyama Itchome or the like). Just go to any of these chain restaurants of Coco Ichibanya and ask for a vegan menu, they will even speak English if they have a vegan menu or if not they will simply say no or don’t understand your question at all (because yeah they don’t even speak English).