I didn't find anything for this season, but the Chicago White Sox had the fewest triples by a team before the All-Star Game in 2024, with 5 triples.

Interpreted as:

team low triples before all star game this season

	TEAM	3B	G	W%	W	L	AB	R	H	2B	HR	RBI	BB	HBP	SO	SB	CS	PA	TB	XBH	SH	SF	IBB	GIDP	AVG	OBP	SLG	OPS
1	White Sox	5	98	.276	27	71	3,237	313	713	145	85	298	256	29	843	63	14	3,553	1,123	235	14	17	6	78	.220	.282	.347	.629
2	Braves	9	95	.558	53	42	3,205	407	774	178	109	386	266	35	844	44	13	3,534	1,297	296	2	26	6	60	.241	.304	.405	.709
3	Mets	9	95	.516	49	46	3,275	465	818	176	121	447	306	50	769	69	15	3,655	1,375	306	8	16	7	65	.250	.322	.420	.742
4	Angels	10	96	.427	41	55	3,181	385	747	144	101	358	277	43	803	80	34	3,529	1,214	255	7	21	3	65	.235	.303	.382	.685
5	Nationals	10	97	.454	44	53	3,211	402	768	158	80	378	280	52	747	126	47	3,585	1,186	248	14	28	7	66	.239	.308	.369	.677
6	Mariners	10	98	.531	52	46	3,203	379	701	129	109	362	319	58	1,013	74	18	3,602	1,177	248	7	15	6	62	.219	.300	.367	.667
7	Pirates	11	96	.500	48	48	3,253	402	751	132	99	389	301	31	886	49	10	3,629	1,202	242	12	32	9	63	.231	.299	.370	.669
8	Cardinals	11	96	.521	50	46	3,257	398	799	149	94	374	279	39	807	56	16	3,620	1,252	254	14	31	5	44	.245	.310	.384	.694
9	Guardians	11	95	.611	58	37	3,147	454	765	160	109	424	287	61	695	79	30	3,532	1,274	280	9	28	22	74	.243	.316	.405	.721
10	Rangers	11	96	.479	46	50	3,237	414	779	131	101	397	304	35	745	52	14	3,599	1,235	243	4	19	14	65	.241	.311	.382	.693
11	Astros	11	96	.521	50	46	3,302	457	866	162	110	440	267	39	669	64	16	3,644	1,380	283	6	30	11	72	.262	.322	.418	.740
12	Rays	11	96	.500	48	48	3,199	376	754	150	86	350	311	49	862	95	35	3,587	1,184	247	8	20	6	63	.236	.311	.370	.681
13	Padres	11	99	.505	50	49	3,355	448	876	143	109	432	276	30	662	69	19	3,690	1,368	263	14	15	12	88	.261	.322	.408	.729
14	Athletics	12	98	.378	37	61	3,265	390	740	137	122	376	307	35	949	53	14	3,642	1,267	271	13	22	8	60	.227	.298	.388	.686
15	Marlins	12	96	.344	33	63	3,252	336	760	133	78	327	205	30	798	52	27	3,516	1,151	223	11	18	11	77	.234	.284	.354	.638
16	Yankees	13	98	.592	58	40	3,269	488	806	139	140	469	387	31	786	38	17	3,729	1,391	292	9	33	7	88	.247	.329	.426	.755
17	Blue Jays	13	97	.464	45	52	3,197	394	754	161	87	374	317	42	714	45	9	3,594	1,202	261	10	28	5	83	.236	.311	.376	.687
18	Giants	14	97	.485	47	50	3,277	427	801	164	96	408	320	31	809	31	14	3,668	1,281	274	7	33	11	69	.244	.315	.391	.706
19	Brewers	15	97	.567	55	42	3,296	466	839	153	99	442	363	32	849	133	24	3,714	1,319	267	6	17	8	61	.255	.333	.400	.733
20	Twins	15	96	.563	54	42	3,257	466	827	192	115	443	279	68	757	46	18	3,639	1,394	322	5	30	7	77	.254	.323	.428	.751
21	Rockies	15	97	.351	34	63	3,286	403	801	168	104	385	269	32	927	53	19	3,627	1,311	287	13	27	12	83	.244	.305	.399	.704
22	Phillies	16	96	.646	62	34	3,271	477	846	156	118	456	334	35	804	99	22	3,675	1,388	290	6	29	14	59	.259	.331	.424	.755
23	Cubs	16	98	.480	47	51	3,265	415	768	147	100	391	327	52	865	78	19	3,678	1,247	263	9	25	10	55	.235	.313	.382	.695
24	Dodgers	17	97	.577	56	41	3,313	479	842	176	129	463	374	39	795	60	15	3,760	1,439	322	4	30	19	65	.254	.334	.434	.768
25	Diamondbacks	19	97	.505	49	48	3,280	482	833	167	102	461	324	46	763	59	16	3,706	1,344	288	17	39	12	67	.254	.326	.410	.736