Checkride and Basicmed
Unfortunately if you have never held a valid medical certificate since July 14, 2006, the answer is no.  One of the eligibility requirements for BasicMed is that you must have held a medical certificate after  that date.
§61.89(d) states

The holder of a student pilot certificate may act as pilot in command
of an aircraft without holding a medical certificate issued under part
67 of this chapter provided the student pilot holds a valid U.S.
driver’s license, meets the requirements of §61.23(c)(3), and the
operation is conducted consistent with the requirements of paragraphs
(a) and (b) of this section and the conditions of §61.113(i). Where
the requirements of paragraphs (a) and (b) of this section conflict
with §61.113(i), a student pilot must comply with paragraphs (a) and
(b) of this section.

§61.113(i) required that pilot hold a valid U.S. driver’s license, meet the requirements of §61.23(c)(3), and complies with this section and the conditions listed in §61.23(c)(3).
But §61.23(c)(3)(B) states that a person operating an a/c without a medical certificate under §61.113(i) must have:

At any point after July 14, 2006, have held a medical certificate
issued under part 67 of this chapter.

As such you will have to acquire a medical in this case before you can take the checkride.