I have two independent groups: from one I have drawn a sample of size less than 30 due to its small population, while the second group has sample size 100.
Is it appropriate to use an independent samples t-test in this case?
t-test requires a set of assumptions. It assumes your data is i.i.d. (independent and identically distributed) and comes from a normal distribution. If you care to compare the means of the two groups (and they follow the assumptions), then yes - you can use that test.
As JohnK, you may wish to note if you want to assume equal variance for the two populations (and it is a reasonable rule of thumb to not make that assumption).
Under the assumptions, this test works for small and large sample sizes (and in the case of large sample sizes will approach the z-test).
The two-sample t-test makes no assumption about equal sample sizes. However, if you have $2n$ observations, the best allocation of them is into two groups, each with $n$ observations. This is part of the experimental design; if you already have your observations, then you don’t get to allocate them into groups. (They either got the coronavirus miracle drug or a placebo, for instance, but if you’re designing the study, the best allocation of 10 patients would be into 5 treatment subjects and 5 placebo subjects, at least as far as statistical power goes.)
The two-sample t-test does assume equal variances in the two groups. There is a slight modification called the Welch t-test that accounts for unequal variances. This is the default in R and can be performed in Python, too. Other software like SAS, Stata, and SPSS ought to be able to do the Welch test, though I’ve never done it.
Sure, you do not need equal sample sizes between group. You even not need equal variance, if you use Welch test specification of t-test, which estimate common variance in denominator part. Also you do not need normality of data, just normality of sample means and respectively their difference (if you test H0 as E(Diff)=0)