Greed, ignorance, and hatred are the three sins of Buddhism. If you didn't consider yourself religious you could worship any god you like. But then again, if you weren't religious, why would you do such a thing in the first place?
I've heard that one cool thing about Buddhism is the fact that it has no concept of "sin", it is pretty individual, and it doesn't really push dogma in the sense of saying "you HAVE TO do this" or "you HAVE TO do that".
..So it's up to you, basically.
I'd say, sure, feel free to.
In Japan people pray at both Buddhist and Shinto Temples and shrines, as far as I can tell it dozen`t seem to matter.
Here and in Singapore, I have seen many Buddhists pray at Hindu Temples too.
Following any religion is not like doing Business. Your belief is based on your firm faith but if you have no firm faith, you can jump on going change of religion time and again.
Stick to one religion once and for all. You will gain spiritual progress if you are with one religion. Go on changing and you loose your previous efforts.
Do not ask any ones advise, follow your own conscience.
most greater vehicle Buddhists have no problem with joining in the worship of other religions and many are secretest
in japan many Mahayana Buddhists are are also Shintoist
Neither Shinto nor Buddhism are "me only" religions with the goal of displacing every other religion (preferably by force). Depending upon your point of view either Shinto Kami are just Buddhist Deities with another name or Buddhist Deities are just another Kami so there's no problem. Buddhism has an after-life focus while Shinto focuses on this life, so they're complementary rather than rivals.
Buddhism takes the stance that life is suffering to prepare you for enlightenment.
Shinto takes the stance that life is good and that you should be happy now and that most unhappiness is caused by "spiritual dirt" (demons if you like metaphors or psychological problems for the science minded). Shinto rituals are designed to cleanse the mind and bring you back to the original clean state.
This results in many practitioners of both religions participating in Shinto rituals for life-affirming events (coming of age, marriage, new years, etc.) and having Buddhist funerals and a batsman (Buddhist home alter) for ancestor worship.