This is actually a bit of a wishy-washy subject. It's part of an ongoing evolution of the English language. 10 years ago or so, "Can I make a call?" would be considered wrong in most circumstances and would be part of an ongoing annoyance of adults by younger people who insisted on using the word 'can' to ask for permission. Can does not express the permission to do something. It expresses the ability to do something. (Can you swim? Can you skate? Can you play the piano? Yes, I can play the piano. <-- expressions of ability) For permission you would use the term 'may' (May I watch TV now? May I have a snack? Yes, you may.) ..However, these days this is hardly an issue. The use of 'may' is rare, especially in informal speech and writing, and are often used interchangably to ask for permission. However, this acceptance is ongoing, and some teachers may be a stickler for it. It all depends on how willing they are to see the language change or how stubborn they are to accept changes. Shall, on the other hand, is more suggestive in nature. In literal meaning, it's more like "How about a make a call?" but contains a somewhat more formal feeling to it. So in this case the permission and ability are assumed.

"Can I make a call" is grammatically incorrect (but many americans use it anyway). If by saying "can i make a call" you mean to ask for permission to make a call (for example, you are at somebody's office, you need to make a call, but you don't have a cell phone. So you ask for permission to use office phone). "Shall I make a call" is used when you ask for instructions. (for example, your boss wants you to ask your co-worker to come to work right now. So you ask your boss whether you should call the co-worker right now or not")

"can" and "shall" are called Modal Auxiliaries. These two words have different meaning and uses. 1. [Can I make a call?] - is used for requesting a permission. 2. [Shall I make a call?] - is used for suggestion

"Usage Note: Generations of grammarians and teachers have insisted that can should be used only to express the capacity to do something, and that may must be used to express permission. But children do not use can to ask permission out of a desire to be stubbornly perverse. They have learned it as an idiomatic expression from adults: After you clean your room, you can go outside and play. As part of the spoken language, this use of can is perfectly acceptable. This is especially true for negative questions, such as Can't I have the car tonight? probably because using mayn't instead of can't sounds unnatural. Nevertheless, in more formal usage the distinction between can and may still has many adherents. Only 21 percent of the Usage Panel accepts can instead of may in the sentence Can I take another week to submit the application? The heightened formality of may sometimes highlights the speaker's role in giving permission. You may leave the room when you are finished implies that permission is given by the speaker. You can leave the room when you are finished implies that permission is part of a rule or policy rather than a decision on the speaker's part. For this reason, may sees considerable use in official announcements: Students may pick up the application forms tomorrow." From dictionary.com, entry on 'can'

Can i make a call = MAY I USE YOUR PHONE? (This is more accurate because making a call is the actual dialing of the number). You would also add "It is a local number." Meaning that you will not be dialing a long distance number that usually costs more money. shall i make a call = SHALL I CALL [you]? (Here you are asking a person whether or not you SHOULD or COULD call). The alternative is that you could write instead.