Hi Hani,
sure you can. In fact you can even do it with your smartphone, or for artistical purposes use an old black and white filmcamera. Music videos are not really limited in that regard, there isn´t even a need to show people singing. The most important thing I´d say is Audio, and you´d want to record that separately anyway.
If you do show people singing, you have to synch the files in post production, otherwise it will look awkward (but even that could be characterized as art

). Synching can be done either manual (checking the waveform and clapping at the beginning of every recording would be the cheapest way to do it), by timecode (which usually is rather pricy), or with software - check
Plural Eyes from Red Giant (costs $299 - so if you plan only one music video, it might not be worth it).
Does that help?
I appreciate that you're trying to be helpful here, but there are a few things you have wrong.
First of all, no. You cannot shoot a music video on a phone unless the style of the video calls for that which the phone uniquely produces. There's a huge misconception out there that, simply because artistic skill is paramount, the tools we use essentially do not matter. You're not going to shoot a Beyonce-style music video on a phone! It's simply beyond the technical limitations of the format. You can, however, shoot a "good" music video with a phone, provided that you're working within the limits of the medium. This goes for everything, whether it's grainy 8mm film all the way up to a Super35 digital cinema camera. So you have to ask yourself, what kind of video are you expecting to shoot, and is it feasible with the equipment you have available.
Not really knowing your gear, it's hard for me to really answer this question, though it would seem you're not familiar enough with your own equipment to really answer this for yourself. I know that comes off as critical, and I don't really mean to be bullish, but it's something you're going to have to appreciate before you can really plan your music video.
Now. As for sound. I'm surprised nobody has really corrected this. Music videos are shot with sound as a reference, but the actual sound being used in the edit is the studio recording. Because every performance will be slightly different, using the timecode from the shoot to sync the timecode of the studio recording is frivolous. Rather, the music is synced visually in post. Fortunately this isn't quite as precise as you might imagine, especially when our visual input is further supported by musical timing - though still requires an editorial skillset.
Regardless what you're shooting on, lighting will be most important. If you're going to spend money on rentals, I'd suggest spending it on lighting. Of course, lighting is a skill in and of itself.
EDIT: first of all: any idea how I can link to youtube without having the files embedded here? it was done automatically.
Thank you for your feedback. Written text almost always sounds more rude, than if we set here and talked, so I understand what you mean about bullish - same here - so if this sounds bossy, it isn´t meant to be.
However, I think we definitely have to differentiate between a highly professional music video done by a popular band with an amazing budget, and those that are appearing more and more on youtube and find their way via youtube into private tv channels, etc.. I´m totally with you in regard to the first, but especially for the latter, there is hardly any limitation. Even for big music productions, there are a few examples (even though some are older, but even back then those are really pretty bad despite being very popular), that don´t abide by the general rules.
Like:
(perhaps the best example with 125.000.000 views on youtube - I don´t say it wasn´t filmed professionally, it sure was but other than the shallow DOF, you could reproduce a similar video with an iphone)
(especially in regard to professional lighting)
(a wall lit by the sun would bring a similar result - not the same, but close)
https://youtu.be/a_zKhYXD2u4
https://youtu.be/LB5YkmjalDg
https://youtu.be/H2lbiS1fris (you could go to a amusement park and easily recreate this kind of video)
https://youtu.be/9G4jnaznUoQ (sure professionally lit, but rather "bad" on purpose)
Matt And Kim - Let's Run Away - Official Music Video (shot on gopro only)
Youngsta - Flowing Through my DNA (iPhone 5 Music Video) (a great example shot on iphone)
There are many, many more examples.
my points are:
- with music video (almost) anything is allowed.
- there are a lot of iphone and gopro videos out there (not only music videos) that are much better than other videos shot with arri alexa, red epic and others. It´s not the camera that matters most.
- lighting: yes and no - you can if you love a great result, but you don´t have to.
- if you have the talent for singing, but not for producing a rather expensive music video - go cheap, but go.
I´ve referred to studio sound recording in post #4 btw

Sound is the one and only most important thing