Can Invisalign Fix an Underbite?

While Invisalign MIGHT be able to help, you are asking a lot for a tool not really designed for this.  It is possible that the solution is much more complex, as it is possible the jaw bones themselves do not line up and would need surgery.

The best thing to do is have a detailed exam by an orthodontist, and they can tell for sure what would help best.

Yes. Invisalign can fix an underbite. Most
cases, if severe, will require the use of rubber bands at night to correct an
underbite.

Your picture definitely shows what appears to be a more complex case with a possible upper to lower jaw size discrepancy.  There can be multiple ways of treating this case which might still include Invisalign.  The key is a thorough exam by a dentist who is certified in Invisalign therapy but also has a good relationship with an orthodontist who can treat a more advanced case if a referral is needed.  Actually, my biggest concern is for the potential TMJ (jaw joint problem) that may be developing as indicated by the cracking and tooth/jaw movements you are now seeing.  Most of the time the TMJ problem may need to be treated first to determine its correct position and then to see where the teeth need to be moved to make that position stable and comfortable.  There are various plastic splints that can be used temporarily to stabilize the TMJ.  Make sure the dentist feels comfortable treating TMJ as not all do.

From your picture it appears that you have not only an "underbite" but also a significant difference in upper and lower jaw size and asymmetric jaw development. In many situations of this type some form of jaw surgery is required for correction. The fact that you are having progressive jaw "cracking" is a sign that treatment is best done sooner rather than later ("An ounce of prevention -------- etc.").

Having said that, if you do need surgery, you will probably also need to have some tooth alignment done in preparation for surgery - and Invisalign may be an option for that alignment. I suggest you schedule a consultation with an orthodontist so you can learn more about the nature of your bite imbalance and the approaches to its correction. 

Be careful here.  The picture you have shown and the description "I notice my jaw has been cracking more now" tells my that you have a complex skeletal misalignment problem that necessitates working with a specialist who not only deals with orthodontics, but temperomandibular dysfunction (TMD) as well.  The proper treatment would involve wearing a special mouth orthotic for 4-6 months to get the jaws aligned and functioning properly.  Then you may need widening of the upper jaw (funtional facial orthopedics) (another 4-6 months) and then the possibilities of wearing Invisalign (another year) or braces would exist. 

This way all the bases are covered, the results are highly predictable, and a lifetime correction is possible.  Otherwise shortcuts may result in necessary re-treatment in the future.  I am sorry to be so long winded here, but I personally have retreated hundreds of cases where the TMJ problem was ignored in the original treatment and the result was many years of pain and dysfunction and ultimately re-treatment to solve the neglected problem.  Do see a doctor who has a great deal of experience in both orthodontics and TMJ Dysfunction.  Good luck to you.