Pure formality would strongly discourage any first person phrasing at all, though to what extent that still exists, never mind with the "anybody with a pulse and a clean criminal history" thing that others alluded to as being the present job world (though TEFL stuff can vary with it being the "easy" option for a long term visa for some). I would certainly avoid any on the bulk of the CV/resume, cover letters are a more interesting debate and I would have 50 first person/I sentences before I had a single errant comma or spelling mistake.
Writing to avoid it is not so bad either.
"I would like to express my interest in this job... I hold these qualifications.... I have this experience... I am willing to do these tasks..."
This cover letter is part of an application for [job title as listed, location as well if worthwhile]. It is of great interest this is sent for your consideration.
Most TEFL stuff seems to want a degree in anything vaguely accredited from an English speaking university whenever I go looking so where normally I would spit their qualifications back at them twisted to my stuff this might be less necessary. If you have done any teaching, had apprentices, done any childcare (assuming you are going for that -- some want you to teach businessmen* or later years teenagers) then wind that in there.
*if your degree or whatever is in IT, something medical, something legal, something engineering, .... such that you can bridge a gap where one that spent 3 years contemplating the deeper meaning of Beowulf and Shakespeare would have no clue then play up that a bit. Anything there necessarily also includes the ability to explain it to the clueless, possibly including sales pitches, new software, new hardware, new ways of doing things, new versions with changes and explaining to manglement.
The requirements list a degree, my degree being in. Further to that and as part of roles using it a considerable amount of tuition in [said field] to technical and non technical staff, customers, investors and managers. Taking to the concept the company tasked myself to deal with that for further instances. Ultimately it became natural and quite enjoyable, and could readily be turned to [this particular application].
Whether my is acceptable in the no first person world we could debate, certainly preferable to an(other) I. Considerable amount is a fluff term but this is job application so kind of expected to do it.
These tasks is an interesting one, though necessary.
The previous roles taught the value of adaptability, flexibility and taking initiative to prevent failure and ensure successful outcomes, teaching and communication being part of that. To that end all reasonable accommodations will be taken to ensure in this role has the desired outcome for the clients, the company and myself. With that said there are tasks which will be unable to be undertaken [maybe you don't fancy cleaning bogs in some nowhere, middle of school and thus best to mention that in this].