To do, play する・します

You can omit 'o' in any of the above examples.
Also, you can use 'o' in any of them.
If you omit 'o', it sounds more colloquial.

ジョギングをする is a completely natural Japanese expression.
ジョギングする is preferred at times, presumably because the word ジョギング is long (has many syllables).

Edit:
When a noun has intrinsically some meaning of "action" in it, you can directly append 'suru' to it to make a verb, such as 心配する(to worry), 飲酒する(to drink), and ジョギングする. These are not considered to be colloquial.

However, when a noun doesn't have such meaning, or 'suru' has the meaning of 'to play', you need 'o' before 'suru' (e.g. 鍵をする(to lock), 野球をする(to play baseball)). You can surely omit 'o' in these cases also, but it is a more colloquial usage.

P.S.
日本語を勉強する becomes 日本語の勉強をする if you don't omit 'o'.

Just like Sam says like he feels there should be an を which is eventually a topic marker.
Question is - is jogging in a very usual sense, an object at all? To me, or others who agree to this subject, jogging is necessarily not an object , but an action that also happens to suffice its conditions as a topic. For example:

ジョッギングしましょう。 - Let's go for jogging.
Here jogging is a gerund [verb to jog+ing]- an action, but not an object

Another example I would like to phrase here is:

愛する　－ To love (no usage of を occurs)
愛します－　I love (No occurence of topic marker を takes place)

The particle を "o" is the direct object marker used for verbs.

Whenever you have a verb that acts directly upon the object you connect it using this particle.

野球　を　する yakyuu o suru. To play baseball.
Doing (o) basketball
相撲 を する。 sumou o suru. To play (do) Sumo.
Doing (o) sumo

日本語　を　勉強　します。
nihongo o benkyou shimasu.
To study (o) Japanese

Some other examples
本を開けてください　hon o akete kudasai
please open (o) book

酒を飲む　sake o nomu
drink (o) sake/alcohol