The PDFs and CDFs of Student's t and chi-squared distributions are known
and are displayed in the corresponding Wikipedia articles. The PDFs are
shown in appendixes of many intermediate probability and mathematical
statistics courses. Your specific questions involve use of the quantile
functions (inverse CDFs). In principle, methods of numerical integration
can be used to get probabilities from PDFs. Also, mathematical methods
can be used to invert CDFs for the same purpose.
An important computational difficulty is that the PDFs and CDFs of Student's
t and chi-squared distributions are expressed in terms of gamma functions,
which must also be evaluated by numerical methods.
You are correct that printed tables give results for Student's t distributions
that arise in most practical applications--and for chi-squared distributions
in many practical applications. Statistical software packages (SAS, Minitab,
SPSS, R, and so on) have functions that give a wider range of quantiles
than you will find in tables. Statistical calculators and Excel give useful approximations.
Chapter 26 of Abramowitz and Stegen (PDF file available online) catalogues some rational approximations that
give reasonable accuracy over various ranges of parameters. However, these
are less frequently used nowadays because it is easy to get more accurate
results from software. (Some software functions use carefully vetted rational approximations,
but generally more intricate and accurate ones than you will find in A&S.)