I am aware that IBM, Rigetti, and Google have built some small-scale devices. Which of them are available for access by an undergraduate student? For how long? With how many qubits?
The first cloud device was made available back in 2013. It is a photonic chip at the University of Bristol. Though it is an example of something we could build quantum computers from, it is quite different from the usual 'circuit model' architecture.
Then 2016 brought some devices from IBM. There are 5 qubit processors anyone can use with the Quantum Experience GUI or using a the QISKit SDK. There's also a 16 qubit device that you can use only with QISKit. Larger devices also exist, but they are not publically available.
Then came Rigetti's 19 qubit QPU at the end of last year, which can be used via pyQuil. You have to apply for an appointment to get access for a few hours, but I see no reason why anyone with a serious interest would not have it granted.
Finally, Alibaba and the Chinese Academy of Sciences have teamed up to put an 11 qubit device on the cloud. The interface is via a GUI, and is quite similar to IBM's Quantum Experience.
An undergrad can create an account with the IBM quantum experience. Users have some points that they can use to run simulations of their design on a real quantum computer. You can use five qubits.
I am unaware of any way for someone to use Google's quantum computer unless you get a job with them. Rigetti has a device, but what people use online is a simulator of a quantum computer, meant to assist with writing quantum algorithms, that runs up to 36 qubits. It uses a language called Quil.
There are multiple simulators available. IBM has one as a part of its IBM quantum experience, there is Quirk, and there are multiple "programming" languages like QCL, most of which are available on GitHub or with the creation of an account, so definitely open to an undergrad.
To summarize: if you don't have a quantum computer of your own and wish to try out a circuit on a real quantum computer, your options seem to be limited to IBM's quantum experience, which has five qubits.